Algebra Quiz: Explaining And Justifying Equation Solving Steps
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Explaining And Justifying Equation Solving StepsQuestion 1 of 20

Two students solve x2−5x+6=0x^2-5x+6=0.

Student A:

  1. x2−5x+6=0x^2-5x+6=0
  2. (x−2)(x−3)=0(x-2)(x-3)=0
  3. x−2=0x-2=0 or x−3=0x-3=0

Student B:

  1. x2−5x+6=0x^2-5x+6=0
  2. x(x−5)+6=0x(x-5)+6=0
  3. x=5x=5 or 6=06=0

Which statement correctly explains why Student A's step 3 is valid?

Zero Product Property: if (x−2)(x−3)=0(x-2)(x-3)=0, then x−2=0x-2=0 or x−3=0x-3=0
Distributive Property: if (x−2)(x−3)=0(x-2)(x-3)=0, then x2−5x+6=0x^2-5x+6=0
Addition Property of Equality: add 22 to both sides to get x=2x=2 or x=3x=3
Division Property of Equality: divide both sides by (x−2)(x-2) to get x−3=0x-3=0
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Algebra Quiz

Algebra Quiz: Explaining And Justifying Equation Solving Steps

Practice Explaining And Justifying Equation Solving Steps in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Explaining And Justifying Equation Solving Steps, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Two students solve x2−5x+6=0x^2-5x+6=0.

Student A:

  1. x2−5x+6=0x^2-5x+6=0
  2. (x−2)(x−3)=0(x-2)(x-3)=0
  3. x−2=0x-2=0 or x−3=0x-3=0

Student B:

  1. x2−5x+6=0x^2-5x+6=0
  2. x(x−5)+6=0x(x-5)+6=0
  3. x=5x=5 or 6=06=0

Which statement correctly explains why Student A's step 3 is valid?

  1. Zero Product Property: if (x−2)(x−3)=0(x-2)(x-3)=0, then x−2=0x-2=0 or x−3=0x-3=0 (correct answer)
  2. Distributive Property: if (x−2)(x−3)=0(x-2)(x-3)=0, then x2−5x+6=0x^2-5x+6=0
  3. Addition Property of Equality: add 22 to both sides to get x=2x=2 or x=3x=3
  4. Division Property of Equality: divide both sides by (x−2)(x-2) to get x−3=0x-3=0
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Looking at Student A's step from (x - 2)(x - 3) = 0 to x - 2 = 0 or x - 3 = 0: this uses the Zero Product Property, which states that if a product of factors equals zero, then at least one of the factors must equal zero. This is a special property that applies when we have a product equal to zero—it's different from the properties of equality but equally important in equation solving. We can apply it here because we have two factors multiplied together equaling zero. Choice A correctly identifies the Zero Product Property and explains it properly: if (x - 2)(x - 3) = 0, then either x - 2 = 0 or x - 3 = 0 (or both), which leads to the solutions x = 2 or x = 3. Choice D suggests using the Division Property to divide by (x - 2), but this is dangerous and incorrect! We cannot divide both sides by (x - 2) because it might equal zero (when x = 2). Dividing by zero is undefined and would lose the x = 2 solution. The Zero Product Property is the correct approach for equations where a product equals zero. A complete justification has three parts: (1) What you did ('applied Zero Product Property'), (2) To what ('to the factored form'), (3) Which property justifies it ('Zero Product Property: if ab = 0, then a = 0 or b = 0'). This property is special because it only works when the product equals zero—it wouldn't apply if we had (x - 2)(x - 3) = 5, for example!

Question 2

A student is solving the equation 3(x−4)+2=2x−73(x-4)+2=2x-7 and writes the steps below.

  1. 3(x−4)+2=2x−73(x-4)+2=2x-7
  2. 3x−12+2=2x−73x-12+2=2x-7
  3. 3x−10=2x−73x-10=2x-7
  4. x−10=−7x-10=-7

Which statement correctly justifies the step from line 3 to line 4?

  1. Distributive Property: distribute 33 across (x−4)(x-4).
  2. Addition Property of Equality: add 1010 to both sides.
  3. Subtraction Property of Equality: subtract 2x2x from both sides. (correct answer)
  4. Division Property of Equality: divide both sides by 33.
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 3x - 10 = 2x - 7 to x - 10 = -7: we subtracted 2x from both sides. This is justified by the Subtraction Property of Equality, which states that if a = b, then a - c = b - c for any c. We can see this is valid because left side 3x - 2x - 10 = x - 10 and right side 2x - 2x - 7 = -7—same operation applied to both sides, so equality is preserved! Choice C correctly identifies the property as the Subtraction Property of Equality because subtracting 2x from both sides isolates the variable term while preserving the equation's balance. Choice A names the wrong property: it says Distributive Property, but we're actually subtracting terms from both sides, which is the Subtraction Property. It's easy to confuse Distributive with Subtraction, but remember: Distributive is about expanding expressions like 3(x-4), while Subtraction is about doing the same subtraction to both sides. The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four! When justifying, identify which one applies and always mention 'both sides.'

Question 3

Which property justifies the step from x−14=3\frac{x-1}{4}=3 to x−1=12x-1=12?​

  1. Division Property of Equality
  2. Multiplication Property of Equality (correct answer)
  3. Subtraction Property of Equality
  4. Distributive Property
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from x−14=3\frac{x-1}{4}=3 to x−1=12x-1=12: we multiplied both sides by 4. This is justified by the Multiplication Property of Equality, which states that if we multiply both sides by the same nonzero value, the equality is preserved. We can see this is valid because left side: 4⋅x−14=x−14 \cdot \frac{x-1}{4} = x-1 and right side: 4⋅3=124 \cdot 3 = 12—same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Multiplication Property of Equality because we multiplied both sides by 4 to clear the denominator. Choice A (Division) would mean dividing both sides, choice C (Subtraction) would mean subtracting from both sides, and choice D (Distributive) is about transforming expressions, not maintaining equality between sides. The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four!

Question 4

Which property justifies the step from 5(x+2)=205(x+2)=20 to x+2=4x+2=4?

Step shown: 5(x+2)=20  →  x+2=45(x+2)=20 \;\rightarrow\; x+2=4

  1. Subtraction Property of Equality
  2. Addition Property of Equality
  3. Distributive Property
  4. Division Property of Equality (correct answer)
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 5(x + 2) = 20 to x + 2 = 4: we divided both sides by 5. This is justified by the Division Property of Equality, which states that if we divide both sides by the same nonzero value, the equality is preserved. We can see this is valid because: left side (5(x + 2) ÷ 5 = x + 2) and right side (20 ÷ 5 = 4)—same operation applied to both sides, so equality is preserved! Choice C correctly identifies the property as the Division Property of Equality because we divided both sides by 5 to eliminate the coefficient of the parenthetical expression. Choice A names the wrong property: it says Distributive Property, but we're actually dividing both sides by 5, which is the Division Property. It's easy to confuse the Distributive Property with the Division Property when parentheses are involved, but remember: the Distributive Property transforms one expression (like 5(x + 2) into 5x + 10), while the Division Property of Equality maintains balance between two sides of an equation by dividing both by the same value. Don't confuse properties: The Distributive Property (a(b+c) = ab + ac) transforms one expression. The properties of equality transform both sides of an equation. Other algebraic properties like Commutative (a + b = b + a) and Associative ((a + b) + c = a + (b + c)) also transform expressions. Properties of equality are specifically about maintaining balance in equations—they're the 'do to both sides' rules!

Question 5

What property allows you to go from x5=7\dfrac{x}{5}=7 to x=35x=35?

Step shown: x5=7  →  x=35\dfrac{x}{5}=7 \;\rightarrow\; x=35

  1. Division Property of Equality
  2. Multiplication Property of Equality (correct answer)
  3. Addition Property of Equality
  4. Subtraction Property of Equality
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from x/5 = 7 to x = 35: we multiplied both sides by 5. This is justified by the Multiplication Property of Equality, which states that if we multiply both sides by the same nonzero value, the equality is preserved. We can see this is valid because: left side (x/5 × 5 = x) and right side (7 × 5 = 35)—same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Multiplication Property of Equality because we multiplied both sides by 5 to eliminate the fraction and isolate x. Choice A names the wrong property: it says Division Property, but we're actually multiplying both sides by 5, which is the Multiplication Property. It's easy to think 'we're dealing with division in x/5, so use Division Property,' but remember: to undo division by 5, we MULTIPLY by 5. The Multiplication Property justifies this step! The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four! When justifying, identify which one applies and always mention 'both sides.'

Question 6

Why is this step valid?

Step shown: If x−4=9x-4=9, then x−4+4=9+4x-4+4=9+4.

Which property of equality justifies this step?

  1. Multiplication Property of Equality
  2. Subtraction Property of Equality
  3. Addition Property of Equality (correct answer)
  4. Symmetric Property of Equality
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from x - 4 = 9 to x - 4 + 4 = 9 + 4: we added 4 to both sides. This is justified by the Addition Property of Equality, which states that if we add the same value to both sides of an equation, the equality is preserved. We can see this is valid because we're adding the exact same value (4) to both the left side and the right side—this maintains the balance of the equation! Choice A correctly identifies the property as the Addition Property of Equality because we're adding 4 to both sides to isolate x. Choice D mentions the Symmetric Property of Equality, which states that if a = b, then b = a (we can flip the sides). But that's not what's happening here—we're not flipping sides, we're adding 4 to both sides. The Symmetric Property is about reversing the order of an equation, while the Addition Property is about adding the same value to maintain equality. A complete justification has three parts: (1) What you did ('added 4'), (2) To where ('to both sides'), (3) Which property justifies it ('Addition Property of Equality'). Example: 'Added 4 to both sides using the Addition Property of Equality.' This format ensures you've covered all the bases!

Question 7

A student factors and then solves:

  1. (x−2)(x+5)=0(x-2)(x+5)=0
  2. x−2=0x-2=0 or x+5=0x+5=0

What property allows the student to go from line 1 to line 2?

  1. Distributive Property
  2. Addition Property of Equality
  3. Reflexive Property of Equality
  4. Zero Product Property (correct answer)
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. The step '(x-2)(x+5)=0 → x-2=0 or x+5=0' is justified by the Zero Product Property: this property states that if a product is zero, then at least one factor must be zero. In this step, we set each factor equal to zero, which allows solving for x. This is different from Distributive Property because we're dealing with the condition for zero product, not expansion. Choice C correctly identifies the property as the Zero Product Property because it enables splitting the equation into two simpler equations. Choice A confuses the Zero Product Property with the Distributive Property: the Distributive Property transforms one expression (like (x-2)(x+5) into x² + 3x - 10), while the Zero Product Property maintains the factored form and sets factors to zero. They're both important but serve different purposes in solving! Don't confuse properties: The Distributive Property (a(b+c) = ab + ac) transforms one expression. The properties of equality transform both sides of an equation. Other algebraic properties like Commutative (a + b = b + a) and Associative ((a + b) + c = a + (b + c)) also transform expressions. Properties of equality are specifically about maintaining balance in equations—they're the 'do to both sides' rules!

Question 8

Which property justifies the step from 5x−7=185x-7=18 to 5x=255x=25?​

  1. Multiplication Property of Equality
  2. Subtraction Property of Equality
  3. Addition Property of Equality (correct answer)
  4. Division Property of Equality
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 5x−7=185x-7=18 to 5x=255x=25: we added 7 to both sides. This is justified by the Addition Property of Equality, which states that if we add the same quantity to both sides of an equation, the equality is preserved. We can see this is valid because left side: 5x−7+7=5x5x-7+7 = 5x and right side: 18+7=2518+7 = 25—same operation applied to both sides, so equality is preserved! Choice C correctly identifies the property as the Addition Property of Equality because we added 7 to both sides to isolate the 5x5x term. Choice A (Multiplication) would mean multiplying both sides by something, choice B (Subtraction) would mean subtracting from both sides, and choice D (Division) would mean dividing both sides—none of which happened here. The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four!

Question 9

Which property justifies the step from 3x+5=143x+5=14 to 3x=93x=9?

Step shown: 3x+5=14  →  3x=93x+5=14 \;\rightarrow\; 3x=9

  1. Division Property of Equality
  2. Subtraction Property of Equality (correct answer)
  3. Addition Property of Equality
  4. Distributive Property
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 3x + 5 = 14 to 3x = 9: we subtracted 5 from both sides. This is justified by the Subtraction Property of Equality, which states that if we subtract the same value from both sides of an equation, the equality is preserved. We can see this is valid because: left side (3x + 5 - 5 = 3x) and right side (14 - 5 = 9)—same operation applied to both sides, so equality is preserved! Choice C correctly identifies the property as the Subtraction Property of Equality because we subtracted 5 from both sides to isolate the term with x. Choice A names the wrong property: it says Addition Property, but we're actually subtracting 5 from both sides, which is the Subtraction Property. It's easy to confuse Addition with Subtraction when thinking about 'undoing' operations, but remember: Subtraction Property is about subtracting the same value from both sides, while Addition Property is about adding the same value to both sides. A complete justification has three parts: (1) What you did ('subtracted 5'), (2) To where ('from both sides'), (3) Which property justifies it ('Subtraction Property of Equality'). Example: 'Subtracted 5 from both sides using the Subtraction Property of Equality.' This format ensures you've covered all the bases!

Question 10

In solving 2(x+1)−3(x−2)=72(x + 1) - 3(x - 2) = 7, a student writes: "Step 1: 2x+2−3x+6=72x + 2 - 3x + 6 = 7" and "Step 2: −x+8=7-x + 8 = 7". What mathematical principles justify the transition from Step 1 to Step 2?

  1. The commutative property allows rearranging terms, then combining like terms using the distributive property
  2. The associative property groups like terms together, then the addition property combines coefficients
  3. Like terms can be combined using properties of real numbers: 2x+(−3x)=(2−3)x2x + (-3x) = (2-3)x and 2+6=82 + 6 = 8 (correct answer)
  4. The substitution property replaces multiple terms with single terms while maintaining equation balance
Explanation: Combining like terms relies on the distributive property in reverse: 2x+(−3x)=(2+(−3))x=−x2x + (-3x) = (2 + (-3))x = -x and 2+6=82 + 6 = 8. This uses properties of real number operations to simplify expressions while maintaining equality. Choice A mentions commutative property but misapplies distributive property. Choice B incorrectly describes the process. Choice D is too vague and doesn't specify which mathematical principles are involved.

Question 11

When solving −4x=12-4x = 12, a student divides both sides by −4-4 to get x=−3x = -3. A classmate argues this step is invalid because "you can't divide by a negative number in equations." How should the first student respond?

  1. The step is invalid because negative division introduces extraneous solutions that must be checked separately
  2. The step is invalid because dividing by negatives changes the inequality direction in all equation types
  3. The step is valid only if you first multiply both sides by −1-1 to make the coefficient positive
  4. The step is valid because the division property of equality applies to all nonzero numbers, including negatives (correct answer)
Explanation: When you encounter negative coefficients in linear equations, remember that the division property of equality works with all nonzero numbers. This property states that if you divide both sides of an equation by the same nonzero number, the equation remains balanced. The first student's work is completely correct. Dividing both sides of −4x=12-4x = 12 by −4-4 gives x=12−4=−3x = \frac{12}{-4} = -3. You can verify this: substituting x=−3x = -3 back into the original equation yields −4(−3)=12-4(-3) = 12, which simplifies to 12=1212 = 12 ✓. Let's examine why the other options are incorrect. Choice A is wrong because dividing by negative numbers doesn't create extraneous solutions—that typically happens with operations like squaring both sides or multiplying by expressions containing variables. Choice B confuses equations with inequalities; while dividing inequalities by negative numbers does flip the inequality sign, equations don't have directional signs to flip. Choice C suggests an unnecessary extra step—while multiplying both sides by −1-1 would work, it's not required since you can divide directly by the negative coefficient. Choice D correctly identifies that the division property of equality applies to all nonzero numbers, whether positive or negative. Study tip: Don't let negative coefficients intimidate you in linear equations. The same algebraic properties apply regardless of sign. However, do remember that inequalities are different—they require flipping the inequality symbol when multiplying or dividing by negative numbers.

Question 12

Solve 2(x−3)=102(x-3)=10 and justify each step using properties of equality and algebraic properties.

Which sequence of justifications correctly matches a valid solution path?

Step 1: 2(x−3)=10→2x−6=102(x-3)=10 \rightarrow 2x-6=10 Step 2: 2x−6=10→2x=162x-6=10 \rightarrow 2x=16 Step 3: 2x=16→x=82x=16 \rightarrow x=8

  1. Step 1 Distributive Property; Step 2 Addition Property of Equality; Step 3 Division Property of Equality. (correct answer)
  2. Step 1 Addition Property of Equality; Step 2 Distributive Property; Step 3 Multiplication Property of Equality.
  3. Step 1 Distributive Property; Step 2 Division Property of Equality; Step 3 Addition Property of Equality.
  4. Step 1 Commutative Property; Step 2 Subtraction Property of Equality; Step 3 Distributive Property.
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The Distributive Property (a(b + c) = ab + ac) is different from the properties of equality: it's about transforming one expression into an equivalent form, not about maintaining equality between two sides. But it's used frequently in equation solving—like expanding 2(x - 3) to 2x - 6—and needs to be distinguished from the equality properties that justify working with both sides. Let's solve 2(x-3)=10 with full justification: Starting equation: 2(x-3)=10 (Given). Step 1: Distribute 2 → 2x-6=10 (Justification: Distributive Property because 2*(x-3)=2x-6 expands the left side equivalently). Step 2: Add 6 to both sides → 2x=16 (Justification: Addition Property of Equality because adding 6 to both sides isolates the term with x). Step 3: Divide both sides by 2 → x=8 (Justification: Division Property of Equality because dividing both sides by 2 (nonzero) isolates x). We've constructed a valid argument showing that IF the equation has a solution, it must be x=8. Choice A correctly identifies the justifications as Step 1 Distributive Property, Step 2 Addition Property of Equality, Step 3 Division Property of Equality because each matches the operation: distributing in Step 1, adding to both sides in Step 2, and dividing both sides in Step 3. Choice D identifies the wrong properties: it says Step 1 Commutative, but we're actually distributing, which is the Distributive Property. It's easy to confuse Commutative with Distributive, but remember: Commutative is about order (a + b = b + a), while Distributive is about multiplying over addition. Don't confuse properties: The Distributive Property (a(b+c) = ab + ac) transforms one expression. The properties of equality transform both sides of an equation. Other algebraic properties like Commutative (a + b = b + a) and Associative ((a + b) + c = a + (b + c)) also transform expressions. Properties of equality are specifically about maintaining balance in equations—they're the 'do to both sides' rules!

Question 13

Explain each step in solving 2(x−3)=102(x-3)=10 by naming the property used in each step.

Work shown:

  1. 2(x−3)=102(x-3)=10
  2. 2x−6=102x-6=10
  3. 2x=162x=16
  4. x=8x=8

Which statement correctly explains step 2 (2x−6=10→2x=162x-6=10 \rightarrow 2x=16)?

  1. Subtraction Property of Equality: subtract 66 from both sides
  2. Multiplication Property of Equality: multiply both sides by 66
  3. Addition Property of Equality: add 66 to both sides (correct answer)
  4. Distributive Property: distribute the 22 across (x−3)(x-3)
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Looking at the transformation from 2x - 6 = 10 to 2x = 16: we added 6 to both sides. This is justified by the Addition Property of Equality, which states that if we add the same value to both sides of an equation, the equality is preserved. We can see this is valid because: left side (2x - 6 + 6 = 2x) and right side (10 + 6 = 16)—same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Addition Property of Equality and specifies that we add 6 to both sides because that's exactly what happened to transform 2x - 6 = 10 into 2x = 16. Choice A says Subtraction Property and subtract 6, but we're actually adding 6 to both sides to eliminate the -6 on the left side. It's easy to think 'we're getting rid of -6 so we subtract,' but remember: to eliminate -6, we ADD 6 (because -6 + 6 = 0). The Addition Property is what justifies this step! When checking if a step is valid, ask yourself: 'Did I do the EXACT SAME thing to BOTH sides of the equation?' If yes, and you used one of the properties of equality, the step is valid. If you only operated on one side, or did different things to each side, the step breaks the equation and is invalid. The 'both sides' check catches most errors!

Question 14

Fill in the reasoning: Solve 1x+2=3\dfrac{1}{x}+2=3 and justify the step from line 2 to line 3.

Work shown:

  1. 1x+2=3\dfrac{1}{x}+2=3
  2. 1x=1\dfrac{1}{x}=1
  3. 1=x1=x

What justification is used in going from line 2 to line 3?

  1. Substitution Property of Equality
  2. Multiplication Property of Equality: multiply both sides by xx (correct answer)
  3. Addition Property of Equality: add xx to both sides
  4. Distributive Property
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 1/x = 1 to 1 = x: we multiplied both sides by x. This is justified by the Multiplication Property of Equality, which states that if we multiply both sides by the same nonzero value, the equality is preserved. We can see this is valid because: left side (1/x × x = 1) and right side (1 × x = x)—same operation applied to both sides, so equality is preserved! Note that we're assuming x ≠ 0 since we started with 1/x. Choice B correctly identifies the property as the Multiplication Property of Equality and specifies that we multiply both sides by x, which transforms 1/x = 1 into 1 = x (or equivalently x = 1). Choice C says Addition Property and add x to both sides, but adding x to 1/x = 1 would give us 1/x + x = 1 + x, which doesn't help isolate x. We need to multiply by x to eliminate the fraction, not add x. The Multiplication Property is what allows us to clear fractions by multiplying! When checking if a step is valid, ask yourself: 'Did I do the EXACT SAME thing to BOTH sides of the equation?' If yes, and you used one of the properties of equality, the step is valid. If you only operated on one side, or did different things to each side, the step breaks the equation and is invalid. The 'both sides' check catches most errors!

Question 15

Identify the error in the student work shown for solving x+3=5\sqrt{x+3}=5.

Student work:

  1. x+3=5\sqrt{x+3}=5
  2. x+3=25x+3=25
  3. x=22x=22
  4. 22+3=5\sqrt{22+3}=5 so the solution is x=22x=22

Which statement best explains the error?

  1. There is no error; each step follows from properties of equality and the check is correct. (correct answer)
  2. Step 3 is invalid because subtracting 33 from both sides violates the Subtraction Property of Equality.
  3. Step 2 is invalid because squaring both sides is never allowed with radicals.
  4. Step 4 is incorrect because 22+3=25=5\sqrt{22+3}=\sqrt{25}=5 is false.
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Let's examine the student work: Step 1: √(x + 3) = 5 (given), Step 2: x + 3 = 25 (squared both sides), Step 3: x = 22 (subtracted 3 from both sides), Step 4: √(22 + 3) = √25 = 5 ✓ (verification). Each step is valid! Step 2 uses the property that if a = b, then a² = b² (squaring both sides maintains equality for positive values). Step 3 uses the Subtraction Property of Equality. The check in Step 4 confirms that x = 22 satisfies the original equation. Choice D correctly states there is no error because each step follows from properties of equality and the check verifies the solution is correct. Choice C claims the check √(22 + 3) = √25 = 5 is false, but this is incorrect: √25 = 5 is true! The student correctly verified that when x = 22, the left side of the original equation becomes √(22 + 3) = √25 = 5, which equals the right side. The solution and verification are both correct. When checking if a step is valid, ask yourself: 'Did I do the EXACT SAME thing to BOTH sides of the equation?' If yes, and you used one of the properties of equality, the step is valid. Squaring both sides is a valid operation (though we must be careful about extraneous solutions with radicals), and the student's work here is completely correct!

Question 16

Identify the error in the student work shown for solving 4x−9=34x-9=3.

Student work:

  1. 4x−9=34x-9=3
  2. 4x=124x=12
  3. x=12−4x=12-4
  4. x=8x=8

Which statement best identifies the first incorrect step and why it is invalid?

  1. Step 3 is incorrect: dividing both sides by 44 should give x=124x=\dfrac{12}{4}, not x=12−4x=12-4. (correct answer)
  2. Step 1 is incorrect: the equation must be written as 4x=3−94x=3-9 first.
  3. Step 2 is incorrect: the student should have subtracted 99 from both sides, not added 99.
  4. There is no error; all steps follow from the Division Property of Equality.
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. A common error in equation solving is applying operations to only one side of the equation: if you subtract 5 from the left side, you MUST subtract 5 from the right side too. This is what the properties of equality guarantee—we do the SAME thing to BOTH sides to maintain the equality. Forgetting this 'both sides' rule is how equations get broken! Let's examine the student work: Step 1: 4x - 9 = 3 (given), Step 2: 4x = 12 (correct—added 9 to both sides), Step 3: x = 12 - 4 (ERROR!), Step 4: x = 8. The error occurs at Step 3: the student wrote x = 12 - 4 instead of dividing both sides by 4. The student appears to have subtracted 4 from 12, which violates the Division Property of Equality because to isolate x from 4x = 12, we must divide BOTH sides by 4, giving x = 12/4 = 3. The correct step would be 4x = 12 → x = 3, applying the Division Property of Equality. Choice B correctly identifies the error as occurring in Step 3 and explains that dividing both sides by 4 should give x = 12/4, not x = 12 - 4, because the Division Property requires dividing both sides by the same value. Choice A incorrectly claims Step 2 has an error, but Step 2 is actually correct: the student properly added 9 to both sides using the Addition Property of Equality. Step 2 transforms 4x - 9 = 3 to 4x = 12, which is valid. The error is in Step 3, not Step 2! For error analysis, go through the student work line by line asking: (1) Is each step justified by a property? (2) Was the same operation applied to both sides? (3) Was arithmetic correct? The error will be where one of these fails. Then explain: 'At Step 3, writing x = 12 - 4 instead of x = 12/4 violates the Division Property because we must divide both sides by 4.' Pinpointing the exact step and naming the violated property is key!

Question 17

Two students solve 2(x−3)=4x+22(x-3)=4x+2.

Student A:

  1. 2(x−3)=4x+22(x-3)=4x+2
  2. 2x−6=4x+22x-6=4x+2
  3. −6=2x+2-6=2x+2
  4. −8=2x-8=2x
  5. x=−4x=-4

Student B:

  1. 2(x−3)=4x+22(x-3)=4x+2
  2. 2x−6=4x+22x-6=4x+2
  3. −2x−6=2-2x-6=2
  4. −2x=8-2x=8
  5. x=−4x=-4

Which statement correctly explains why line 3 of Student A (2x−6=4x+2→−6=2x+22x-6=4x+2 \rightarrow -6=2x+2) is valid?

  1. Subtraction Property of Equality: subtract 2x2x from both sides.
  2. Subtraction Property of Equality: subtract 2x2x from both sides and simplify. (correct answer)
  3. Division Property of Equality: divide both sides by 2x2x.
  4. Addition Property of Equality: add 66 to both sides.
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. The step '2x - 6 = 4x + 2 → -6 = 2x + 2' is justified by the Subtraction Property: this property states that we can subtract the same value from both sides. In this step, we subtracted 2x from both sides, which maintains equality. This is different from adding because we're removing 2x terms. Choice C correctly identifies the property as the Subtraction Property of Equality because it specifies subtracting 2x from both sides and simplifying, which accurately describes the transformation. Choice A names the wrong property: it says subtract 2x from both sides, but omits the simplification note, though more importantly, Choice C includes it for clarity. It's easy to confuse similar descriptions, but remember: Subtraction Property is about subtracting the same value from both sides. A complete justification has three parts: (1) What you did ('subtracted 5,' 'divided by 3'), (2) To where ('from both sides,' 'both sides by'), (3) Which property justifies it ('Subtraction Property of Equality'). Example: 'Subtracted 5 from both sides using the Subtraction Property of Equality.' This format ensures you've covered all the bases!

Question 18

Two students solve x2−5x+6=0x^2-5x+6=0.

Student A:

  1. x2−5x+6=0x^2-5x+6=0
  2. (x−2)(x−3)=0(x-2)(x-3)=0
  3. x=2x=2 or x=3x=3

Student B:

  1. x2−5x+6=0x^2-5x+6=0
  2. x(x−5)+6=0x(x-5)+6=0
  3. x=0x=0 or x=5x=5 or x=6x=6

Which statement is correct?​

  1. Student A is correct because factoring and then applying the Zero Product Property gives x=2x=2 or x=3x=3. (correct answer)
  2. Student B is correct because rewriting as x(x−5)+6=0x(x-5)+6=0 allows the Zero Product Property immediately.
  3. Both students are correct because any factoring method leads to the same three solutions.
  4. Neither student is correct because quadratics cannot be solved using the Zero Product Property.
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Let's examine both students' work: Student A correctly factors x2−5x+6=(x−2)(x−3)x^2-5x+6=(x-2)(x-3) and applies the Zero Product Property to get x=2x=2 or x=3x=3. We can verify: when x=2x=2: (2)2−5(2)+6=4−10+6=0(2)^2-5(2)+6 = 4-10+6 = 0 ✓, and when x=3x=3: (3)2−5(3)+6=9−15+6=0(3)^2-5(3)+6 = 9-15+6 = 0 ✓. Student B incorrectly attempts to factor by writing x(x−5)+6=0x(x-5)+6=0, but this doesn't equal the original expression because x(x−5)+6=x2−5x+6x(x-5)+6 = x^2-5x+6, which is just the original expression, not a factored form. The Zero Product Property only applies when we have a product equal to zero, like (x−2)(x−3)=0(x-2)(x-3)=0, not a sum like x(x−5)+6=0x(x-5)+6=0. Choice A correctly identifies that Student A is correct because proper factoring followed by the Zero Product Property gives the two solutions x=2x=2 or x=3x=3. Choice B incorrectly supports Student B's flawed approach, choice C wrongly claims both are correct, and choice D incorrectly states quadratics can't be solved using Zero Product Property. For error analysis, go through the student work line by line asking: (1) Is each step justified by a property? (2) Was the same operation applied to both sides? (3) Was arithmetic correct? The error will be where one of these fails!

Question 19

While solving 2x+63=8\frac{2x + 6}{3} = 8, a student multiplies both sides by 3 to get 2x+6=242x + 6 = 24. Which statement best explains why this step is valid?

  1. Multiplying by 3 eliminates fractions, which always simplifies equations without changing solutions
  2. The multiplication property of equality states that multiplying both sides by the same nonzero number preserves equality (correct answer)
  3. The inverse operation property requires multiplying by 3 to cancel the division by 3
  4. The distributive property allows multiplying the numerator by 3 while maintaining the equation balance
Explanation: The multiplication property of equality justifies this step: if a=ba = b, then ca=cbca = cb for any nonzero number cc. Since 3 ≠ 0, multiplying both sides by 3 preserves the equality. Choice A is incorrect because eliminating fractions doesn't always preserve solutions (e.g., if multiplying by zero). Choice C mentions inverse operations but doesn't explain why the equality is preserved. Choice D misapplies the distributive property.

Question 20

Solve 2(3x−1)=102(3x-1)=10 and justify each step using properties of equality and algebraic properties. Which option gives a valid sequence of steps with correct justifications?​

  1. Step 1: 2(3x−1)=102(3x-1)=10 → 6x−2=106x-2=10 (Distributive Property) Step 2: 6x−2=106x-2=10 → 6x=126x=12 (Addition Property of Equality) Step 3: 6x=126x=12 → x=2x=2 (Division Property of Equality) (correct answer)
  2. Step 1: 2(3x−1)=102(3x-1)=10 → 3x−1=53x-1=5 (Division Property of Equality) Step 2: 3x−1=53x-1=5 → 3x=43x=4 (Subtraction Property of Equality) Step 3: 3x=43x=4 → x=43x=\frac{4}{3} (Multiplication Property of Equality)
  3. Step 1: 2(3x−1)=102(3x-1)=10 → 6x−1=106x-1=10 (Distributive Property) Step 2: 6x−1=106x-1=10 → 6x=96x=9 (Addition Property of Equality) Step 3: 6x=96x=9 → x=32x=\frac{3}{2} (Division Property of Equality)
  4. Step 1: 2(3x−1)=102(3x-1)=10 → 6x−2=106x-2=10 (Commutative Property) Step 2: 6x−2=106x-2=10 → x−2=10x-2=10 (Division Property of Equality) Step 3: x−2=10x-2=10 → x=12x=12 (Addition Property of Equality)
Explanation: This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Let's solve 2(3x−1)=102(3x-1)=10 with full justification: Starting equation: 2(3x−1)=102(3x-1)=10 (Given). Step 1: Apply Distributive Property to expand 2(3x−1)2(3x-1) → 6x−2=106x-2=10 (Justification: Distributive Property because 2(3x−1)=2(3x)+2(−1)=6x−22(3x-1) = 2(3x) + 2(-1) = 6x - 2). Step 2: Add 2 to both sides → 6x=126x=12 (Justification: Addition Property of Equality because we added 2 to both sides). Step 3: Divide both sides by 6 → x=2x=2 (Justification: Division Property of Equality because we divided both sides by 6). We've constructed a valid argument showing that IF the equation has a solution, it must be 2. Choice A correctly identifies all steps and properties: Distributive Property for expanding, Addition Property for adding 2 to both sides, and Division Property for dividing both sides by 6. Choice B has arithmetic errors in step 2 (should add 1, not subtract), choice C distributes incorrectly (2(3x−1)≠6x−12(3x-1) ≠ 6x-1), and choice D misuses the Commutative Property and makes nonsensical steps. A complete justification has three parts: (1) What you did ('subtracted 5,' 'divided by 3'), (2) To where ('from both sides,' 'both sides by'), (3) Which property justifies it ('Subtraction Property of Equality'). This format ensures you've covered all the bases!