PSAT Math Flashcards: Equations With One Variable

What this deck covers

This deck focuses on Equations With One Variable, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the solution to $\frac{2x+1}{5}=3$?

Answer: $x=7$. Multiply by $5$ to get $2x+1=15$, then solve for $x$.

Flashcard 2: What is the solution set of $3(x-2)=3x-6$?

Answer: Infinitely many solutions. Expanding gives $3x-6=3x-6$, always true for any $x$.

Flashcard 3: What property justifies adding the same number to both sides of $x-7=12$?

Answer: Addition Property of Equality. Allows adding the same value to both sides to maintain equality.

Flashcard 4: What is the result type if solving simplifies to $0=5$?

Answer: No solution. A false statement means no value satisfies the equation.

Flashcard 5: Identify the solution to rac{x}{5}+2=9.

Answer: $x=35$. Subtract 2: rac{x}{5}=7, then multiply by 5.

Flashcard 6: What is the first step to solve rac{x}{3}+rac{x}{2}=5 efficiently?

Answer: Multiply both sides by $6$. Eliminates fractions by finding LCD of 3 and 2.

Flashcard 7: What is the solution to $\frac{2x+1}{5}=3$?

Answer: $x=7$. Multiply by 5, subtract 1, divide by 2: $14÷2=7$.

Flashcard 8: Identify the solution to $2x+5=3x-1$.

Answer: $x=6$. Subtract $2x$ and add 1 to get $x=6$.

Flashcard 9: What is the solution to $x+8=19$?

Answer: $x=11$. Subtract $8$ from both sides to isolate $x$.

Flashcard 10: What property justifies multiplying both sides of $\frac{x}{5}=9$ by $5$?

Answer: Multiplication Property of Equality. Allows multiplying both sides by the same non-zero value.

Flashcard 11: What is the solution to $4(x-2)=20$?

Answer: $x=7$. Divide by $4$ to get $x-2=5$, then add $2$: $x = 7$.

Flashcard 12: What does it mean for two expressions to be equivalent for all $x$?

Answer: They have the same value for every $x$ in the domain. Equivalent expressions produce identical outputs for all inputs.

Flashcard 13: What is the solution to $x^2=49$?

Answer: $x=7$ or $x=-7$. Take square root of both sides: $x=±7$.

Flashcard 14: What is the solution to the equation rac{x}{4}=9?

Answer: $x=36$. Multiply both sides by 4: $x = 9 × 4 = 36$.

Flashcard 15: What is the result of dividing both sides of an equation by $0$?

Answer: It is undefined and not allowed. Division by zero is mathematically undefined.

Flashcard 16: What property justifies multiplying both sides of an equation by the same nonzero number?

Answer: Multiplication Property of Equality. Allows multiplying both sides by same nonzero value to maintain equality.

Flashcard 17: What is the solution of $2x + 3 = 2x - 5$?

Answer: No solution. Same variable terms cancel, leaving $3 = -5$, which is false.

Flashcard 18: What is the solution to $2x+5=2x-1$?

Answer: No solution. Same $x$ coefficient on both sides with different constants.

Flashcard 19: What is the correct classification if an equation simplifies to $0=0$?

Answer: Infinitely many solutions. A true statement like $0=0$ means all values work.

Flashcard 20: Which value is excluded from the domain of rac{1}{x-4} when solving equations?

Answer: $x\ne 4$. Division by zero occurs when denominator equals zero: $x - 4 = 0$.

Flashcard 21: What is the solution to $\frac{3}{4}x=18$?

Answer: $x=24$. Multiply by $\frac{4}{3}$: $x = 18 × \frac{4}{3} = 24$.

Flashcard 22: What is the solution to $\frac{x-2}{3}=4$?

Answer: $x=14$. Multiply both sides by 3, then add 2: $12+2=14$.

Flashcard 23: What is the solution set of $|x-3|=5$?

Answer: $x=8$ or $x=-2$. Absolute value equation: $x - 3 = 5$ or $x - 3 = -5$.

Flashcard 24: What is the solution of $0.5x + 3 = 11$?

Answer: $x = 16$. Subtract 3, then divide by 0.5: $8 ÷ 0.5 = 16$.

Flashcard 25: What is the solution to rac{x}{6}=4?

Answer: $x=24$. Multiply both sides by 6: $4×6=24$.

Flashcard 26: What is the correct classification if an equation simplifies to $0=5$?

Answer: No solution. A false statement like $0=5$ means no value works.

Flashcard 27: What is the distributive property written with a variable?

Answer: $a(b+c)=ab+ac$. Multiply $a$ by each term inside the parentheses.

Flashcard 28: What is the solution of $x^2 = 49$?

Answer: $x = 7$ or $x = -7$. Take square root of both sides: $sqrt{49} = ±7$.

Flashcard 29: What is the solution to $3(x-1)+2=2x+5$?

Answer: $x=6$. Expand: $3x - 3 + 2 = 2x + 5$, simplify to $3x - 1 = 2x + 5$, so $x = 6$.

Flashcard 30: What is the solution to $3(x-1)=3x-3$?

Answer: All real numbers. Distributing gives $3x-3=3x-3$, an identity true for all $x$.

Flashcard 31: What is the solution to $7-3x=1$?

Answer: $x=2$. Subtract $7$, divide by $-3$: $-3x = -6$, so $x = 2$.

Flashcard 32: What is the solution to $\frac{x}{6}=-4$?

Answer: $x=-24$. Multiply both sides by $6$ to isolate $x$.

Flashcard 33: What is $x$ if $\frac{x}{5}-3=1$?

Answer: $x=20$. Add $3$ to get $\frac{x}{5}=4$, then multiply by $5$.

Flashcard 34: Identify the solution set of $3(x-2)=3x+1$.

Answer: No solution. Expands to $3x-6=3x+1$, simplifies to $-6=1$.

Flashcard 35: What property justifies adding the same number to both sides of $a=b$?

Answer: Addition Property of Equality. If $a=b$, then $a+c=b+c$ for any number $c$.

Flashcard 36: What is the solution to the equation $2(x+3)=18$?

Answer: $x=6$. Divide by 2, then subtract 3: $x + 3 = 9$, so $x = 6$.

Flashcard 37: What is the solution to $x+9=2$?

Answer: $x=-7$. Subtract $9$ from both sides: $x = 2 - 9 = -7$.

Flashcard 38: What does it mean for $x=a$ to be a solution of an equation in one variable?

Answer: $x=a$ makes both sides equal when substituted. Substituting $a$ for $x$ makes the equation true.

Flashcard 39: What is the solution to the equation rac{x-5}{3}=4?

Answer: $x=17$. Multiply by 3, then add 5: $x - 5 = 12$, so $x = 17$.

Flashcard 40: What is the solution to the equation $12-3(x-2)=6$?

Answer: $x=4$. Distribute: $12 - 3x + 6 = 6$, so $18 - 3x = 6$, thus $x = 4$.

Flashcard 41: What is the solution to $2x+4=2x-1$?

Answer: No solution. Subtracting $2x$ gives $4=-1$, which is false.

Flashcard 42: What is the solution to $x+7=19$?

Answer: $x=12$. Subtract 7 from both sides: $19-7=12$.

Flashcard 43: What is the solution of $3x - 5 = 16$?

Answer: $x = 7$. Add 5 to both sides, then divide by 3: $21 ÷ 3 = 7$.

Flashcard 44: What conclusion follows if solving gives $0=5$?

Answer: No solution. A false statement means the equation has no valid solutions.

Flashcard 45: What is the solution to $5x=35$?

Answer: $x=7$. Divide both sides by $5$ to isolate $x$.

Flashcard 46: What is the inverse operation of multiplying by $k$ (with $k\ne 0$), as in $kx$?

Answer: Divide by $k$. Division undoes multiplication when $k\ne 0$.

Flashcard 47: What is the solution to $0.5x+2=7$?

Answer: $x=10$. Subtract 2: $0.5x=5$, then divide by 0.5 to get $x=10$.

Flashcard 48: What is the solution of $x^2 - 9 = 0$?

Answer: $x = 3$ or $x = -3$. Factor: $(x + 3)(x - 3) = 0$, so $x = ±3$.

Flashcard 49: What is the solution of rac{x}{4}=9?

Answer: $x=36$. Multiply both sides by $4$: $x=9×4$.

Flashcard 50: What is the solution to $0.5x+1=4$?

Answer: $x=6$. Subtract 1, then multiply by 2: $0.5x = 3$, so $x = 6$.

Flashcard 51: What is the solution of $3x-4=11$?

Answer: $x=5$. Add $4$ to both sides, then divide by $3$.

Flashcard 52: What is $x$ if $0.2x=6$?

Answer: $x=30$. Divide both sides by $0.2$: $x = 6 ÷ 0.2 = 30$.

Flashcard 53: Identify the solution set of $2(x+1)=2x+2$.

Answer: Infinitely many solutions. Expands to $2x+2=2x+2$, always true.

Flashcard 54: Identify the solution to rac{2x-1}{3}=5.

Answer: $x=8$. Multiply by 3: $2x-1=15$, add 1, divide by 2.

Flashcard 55: What is the solution of $4x+3=2x+15$?

Answer: $x=6$. Subtract $2x$ and $3$ from both sides: $2x=12$.

Flashcard 56: What is the solution to $2(x+5)=18$?

Answer: $x=4$. Divide by 2, then subtract 5: $9-5=4$.

Flashcard 57: What is the solution to $6-3(x-2)=0$?

Answer: $x=4$. Distribute: $6-3x+6=0$, so $12-3x=0$, thus $x=4$.

Flashcard 58: What is the solution to $x+8=19$?

Answer: $x=11$. Subtract 8 from both sides: $19-8=11$.

Flashcard 59: What operation property justifies multiplying both sides of $a=b$ by the same nonzero number?

Answer: Multiplication Property of Equality. States that if $a=b$, then $ac=bc$ for any nonzero number $c$.

Flashcard 60: What is the solution to the equation $(x-3)^2=16$?

Answer: $x=7$ or $x=-1$. Take square root: $x - 3 = ±4$, so $x = 7$ or $x = -1$.

Flashcard 61: What is the solution to the equation rac{3}{x}=6?

Answer: x=rac{1}{2}. Cross multiply: $3 = 6x$, so x = rac{3}{6} = rac{1}{2}.

Flashcard 62: What is the solution to $|2x+1|=7$?

Answer: $x=3$ or $x=-4$. The expression $2x+1$ equals $7$ or $-7$.

Flashcard 63: What is the rule for clearing a denominator in rac{x}{5}=7?

Answer: Multiply both sides by $5$. Eliminates the denominator to isolate $x$.

Flashcard 64: What is the solution of $2(x+3)=2x+5$?

Answer: No solution. Distributing gives $2x+6=2x+5$, which simplifies to $6=5$.

Flashcard 65: What is the solution to $|x-3|=5$?

Answer: $x=8$ or $x=-2$. The expression $x-3$ equals $5$ or $-5$.

Flashcard 66: What is $x$ if $5x=35$?

Answer: $x=7$. Divide both sides by $5$: $x = 35 ÷ 5 = 7$.

Flashcard 67: What is the solution to rac{2x+1}{5}=3?

Answer: $x=7$. Multiply by 5: $2x+1=15$, so $2x=14$, thus $x=7$.

Flashcard 68: What is the solution of rac{x}{4} = 9?

Answer: $x = 36$. Multiply both sides by 4: $9 × 4 = 36$.

Flashcard 69: What is the solution to $4x-3=2x+9$?

Answer: $x=6$. Subtract $2x$ from both sides: $2x - 3 = 9$, so $x = 6$.

Flashcard 70: What is the solution to $x-3=2x+5$?

Answer: $x=-8$. Subtract $x$ and add 3: $-3=x+5$, so $x=-8$.

Flashcard 71: What is the solution to $5x=35$?

Answer: $x=7$. Divide both sides by $5$: $x = 35 ÷ 5 = 7$.

Flashcard 72: What is the solution set of $3(x-1)=3x+2$?

Answer: No solution. Expanding gives $3x - 3 = 3x + 2$, which simplifies to $-3 = 2$ (false).

Flashcard 73: What is the solution set of $4(x-1)=4x-4$?

Answer: Infinitely many solutions. Distributing gives $4x-4=4x-4$, which is always true.

Flashcard 74: What is $x$ if $0.2x+1=3$?

Answer: $x=10$. Subtract $1$ to get $0.2x=2$, divide by $0.2$.

Flashcard 75: What is the solution to $0.2x=6$?

Answer: $x=30$. Divide both sides by $0.2$: $x = 6 ÷ 0.2 = 30$.

Flashcard 76: What is $x$ if $\frac{x}{4}=-3$?

Answer: $x=-12$. Multiply both sides by $4$: $x = -3 × 4 = -12$.

Flashcard 77: What is the solution to $4x+3=2x+15$?

Answer: $x=6$. Subtract $2x$ and $3$ from both sides, then divide by $2$.

Flashcard 78: What is the solution of $7-2x=1$?

Answer: $x=3$. Subtract $7$: $-2x=-6$, then divide by $-2$.

Flashcard 79: What is the solution to $7-2x=1$?

Answer: $x=3$. Subtract 7, then divide by $-2$: $-6÷(-2)=3$.

Flashcard 80: What is the solution to $0.2x=8$?

Answer: $x=40$. Divide both sides by 0.2: $8÷0.2=40$.

Flashcard 81: What property justifies adding the same number to both sides of $a=b$?

Answer: Addition property of equality. This property maintains equality when adding to both sides.

Flashcard 82: What is the solution to the equation $3x-5=16$?

Answer: $x=7$. Add 5 to both sides, then divide by 3: $3x = 21$, so $x = 7$.

Flashcard 83: What is the solution to the equation $4x+6=2x+18$?

Answer: $x=6$. Subtract $2x$ and 6 from both sides: $2x = 12$, so $x = 6$.

Flashcard 84: What is the solution to $2x+5=17$?

Answer: $x=6$. Subtract $5$, then divide by $2$: $2x = 12$, so $x = 6$.

Flashcard 85: What is the solution set of $3(x-1)=3x+2$?

Answer: No solution. Distributing gives $3x-3=3x+2$, which simplifies to $-3=2$.

Flashcard 86: What is the solution of $5x = 35$?

Answer: $x = 7$. Divide both sides by 5: $35 ÷ 5 = 7$.

Flashcard 87: What is $x$ if $\frac{2x+1}{5}=3$?

Answer: $x=7$. Multiply by $5$ to get $2x+1=15$, then solve.

Flashcard 88: What is the solution set of $2(x+1)=2x+2$?

Answer: All real numbers. Expanding gives $2x + 2 = 2x + 2$, which is always true.

Flashcard 89: What is $x$ if $|x-3|=7$?

Answer: $x=10$ or $x=-4$. The absolute value equation gives $x-3=7$ or $x-3=-7$.

Flashcard 90: What conclusion follows if solving gives $0=0$?

Answer: Infinitely many solutions. A true statement means all values of $x$ satisfy the equation.

Flashcard 91: What is the solution set of $4(x-1)=4x-4$?

Answer: Infinitely many solutions. Expanding gives $4x-4=4x-4$, which is always true.

Flashcard 92: What is the solution to $0.5x=8$?

Answer: $x=16$. Divide both sides by 0.5: $8÷0.5=16$.

Flashcard 93: What is the solution to $|x-3|=5$?

Answer: $x=8$ or $x=-2$. $x-3=5$ or $x-3=-5$, giving two solutions.

Flashcard 94: What property justifies adding the same number to both sides of $x-7=12$?

Answer: Addition Property of Equality. Allows adding the same value to both sides to maintain equality.

Flashcard 95: What is the solution to $x^2=49$?

Answer: $x=7$ or $x=-7$. Take square root of both sides: $\sqrt{49}=±7$.

Flashcard 96: What is the solution of $5x=35$?

Answer: $x=7$. Divide both sides by $5$: $35÷5=7$.

Flashcard 97: What is the solution to $\frac{x}{4}=-3$?

Answer: $x=-12$. Multiply both sides by 4: $-3×4=-12$.

Flashcard 98: What is the solution to $2(x+3)=x+11$?

Answer: $x=5$. Expand left side: $2x + 6 = x + 11$, then $x = 5$.

Flashcard 99: What is $x$ if $5-2(3x-1)=1$?

Answer: $x=1$. Distribute: $5-6x+2=1$, then $7-6x=1$, so $x=1$.

Flashcard 100: What is the solution of $|x-3|=5$?

Answer: $x=8$ or $x=-2$. $x-3=5$ or $x-3=-5$, giving two solutions.