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ISEE Upper Level Quantitative : Equations

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Example Questions

Example Question #681 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

$\displaystyle x(2x-9)= 35$

Possible Answers:

$x =22 \textrm{ or } x = 3 5$ $\displaystyle x =22 \textrm{ or } x = 3 5$

$x =-5 \textrm{ or } x = 3 \frac{1}{2}$ $\displaystyle x =-5 \textrm{ or } x = 3 \frac{1}{2}$

$x =-7 \textrm{ or } x = 2 \frac{1}{2}$ $\displaystyle x =-7 \textrm{ or } x = 2 \frac{1}{2}$

$x = -2 \frac{1}{2} \textrm{ or }x =7$ $\displaystyle x = -2 \frac{1}{2} \textrm{ or }x =7$

$x = -3 \frac{1}{2} \textrm{ or }x =5$ $\displaystyle x = -3 \frac{1}{2} \textrm{ or }x =5$

Correct answer:

$x = -2 \frac{1}{2} \textrm{ or }x =7$ $\displaystyle x = -2 \frac{1}{2} \textrm{ or }x =7$

Explanation:

First, rewrite the quadratic equation in standard form by distributing the $\displaystyle x$ through the product on the left and collecting all of the terms on the left side:

$\displaystyle x(2x-9)= 35$

$x \cdot 2x-x \cdot 9= 35$ $\displaystyle x \cdot 2x-x \cdot 9= 35$

$2x ^{2}-9x= 35$ $\displaystyle 2x ^{2}-9x= 35$

$2x ^{2}-9x-35= 35 -35$ $\displaystyle 2x ^{2}-9x-35= 35 -35$

$2x ^{2}-9x-35= 0$ $\displaystyle 2x ^{2}-9x-35= 0$

Use the $\displaystyle ac$ method to factor the quadratic expression $2x ^{2}-9x-35$ $\displaystyle 2x ^{2}-9x-35$; we are looking to split the linear term by finding two integers whose sum is $\displaystyle -9$ and whose product is $\displaystyle 2 (-35) = -70$. These integers are -14,5 $\displaystyle -14,5$, so:

$\left (2x ^{2}-14x \right ) + \left ( 5x-35 \right ) = 0$ $\displaystyle \left (2x ^{2}-14x \right ) + \left ( 5x-35 \right ) = 0$

$2x \left (x-7 \right ) + 5\left (x-7 \right ) = 0$ $\displaystyle 2x \left (x-7 \right ) + 5\left (x-7 \right ) = 0$

$\left (2x+ 5 \right ) \left (x-7 \right ) = 0$ $\displaystyle \left (2x+ 5 \right ) \left (x-7 \right ) = 0$

Set each expression equal to 0 and solve:

$\displaystyle 2x+5 = 0$

2x=-5 $\displaystyle 2x=-5$

$x = - \frac{5}{2} = -2 \frac{1}{2}$ $\displaystyle x = - \frac{5}{2} = -2 \frac{1}{2}$

x-7 = 0 $\displaystyle x-7 = 0$

x=7 $\displaystyle x=7$

The solution set is $\left \{ -2 \frac{1}{2}, 7\right \}$ $\displaystyle \left \{ -2 \frac{1}{2}, 7\right \}$.

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Example Question #12 : Equations

Consider the line of the equation $\displaystyle 0.9x + 0.7y = 100$

Which is the greater quantity?

(a) The $\displaystyle x$-coordinate of the $\displaystyle x$-intercept

(b) The $\displaystyle y$-coordinate of the $\displaystyle y$-intercept

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(b) is greater

Explanation:

(a) To find the $\displaystyle x$-coordinate of the $\displaystyle x$-intercept, substitute y = 0 $\displaystyle y = 0$:

$\displaystyle 0.9x + 0.7y = 100$

$0.9x + 0.7 \cdot 0 = 100$ $\displaystyle 0.9x + 0.7 \cdot 0 = 100$

$\displaystyle 0.9x + 0 = 100$

$\displaystyle 0.9x = 100$

$0.9x \div 0.9 = 100 \div 0.9$ $\displaystyle 0.9x \div 0.9 = 100 \div 0.9$

$x = 100 \div 0.9 = 1,000 \div 9 = 111 \frac {1}{9}$ $\displaystyle x = 100 \div 0.9 = 1,000 \div 9 = 111 \frac {1}{9}$

(b) To find the $\displaystyle y$-coordinate of the $\displaystyle y$-intercept, substitute x = 0 $\displaystyle x = 0$:

$\displaystyle 0.9x + 0.7y = 100$

$0.9 \cdot 0 + 0.7y = 100$ $\displaystyle 0.9 \cdot 0 + 0.7y = 100$

$\displaystyle 0 + 0.7y = 100$

$\displaystyle 0.7y = 100$

$0.7y \div 0.7 = 100 \div 0.7$ $\displaystyle 0.7y \div 0.7 = 100 \div 0.7$

$y =100 \div 0.7 = 1,000 \div 7 = 142\frac {6} {7}$ $\displaystyle y =100 \div 0.7 = 1,000 \div 7 = 142\frac {6} {7}$

(b) is the greater quantity.

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Example Question #11 : How To Find The Solution To An Equation

$\displaystyle | 50 - t | = 30$

Which is the greater quantity?

(a) $\displaystyle t$

(b) 0

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

$\displaystyle | 50 - t | = 30$ can be rewritten as a compound statement:

$\displaystyle 50 - t = 30$ or $\displaystyle 50 - t = -30$

Solve both:

$\displaystyle 50 - t = 30$

$\displaystyle 50 - t - 50 = 30 - 50$

$\displaystyle -t = -20$

t = 20 $\displaystyle t = 20$

$\displaystyle 50 - t = -30$

$\displaystyle 50 - t -50 = -30-50$

$\displaystyle -t = -80$

t = 80 $\displaystyle t = 80$

Either way, t > 0 $\displaystyle t > 0$, so (a) is the greater quantity

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Example Question #12 : Equations

Consider the line of the equation $\frac{1}{99}x + \frac{1}{101}y = 1$ $\displaystyle \frac{1}{99}x + \frac{1}{101}y = 1$

Which is the greater quantity?

(a) The $\displaystyle x$-coordinate of the $\displaystyle x$-intercept

(b) The $\displaystyle y$-coordinate of the $\displaystyle y$-intercept

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

(a) To find the $\displaystyle x$-coordinate of the $\displaystyle x$-intercept, substitute y = 0 $\displaystyle y = 0$:

$\frac{1}{99}x + \frac{1}{101}y = 1$ $\displaystyle \frac{1}{99}x + \frac{1}{101}y = 1$

$\frac{1}{99}x + \frac{1}{101} \cdot 0 = 1$ $\displaystyle \frac{1}{99}x + \frac{1}{101} \cdot 0 = 1$

$\frac{1}{99}x + 0 = 1$ $\displaystyle \frac{1}{99}x + 0 = 1$

$\frac{1}{99}x = 1$ $\displaystyle \frac{1}{99}x = 1$

$\frac{1}{99}x \cdot 99= 1\cdot 99$ $\displaystyle \frac{1}{99}x \cdot 99= 1\cdot 99$

x= 99 $\displaystyle x= 99$

(b) To find the $\displaystyle y$-coordinate of the $\displaystyle y$-intercept, substitute x = 0 $\displaystyle x = 0$:

$\frac{1}{99}x + \frac{1}{101}y = 1$ $\displaystyle \frac{1}{99}x + \frac{1}{101}y = 1$

$\frac{1}{99} \cdot 0 + \frac{1}{101}y = 1$ $\displaystyle \frac{1}{99} \cdot 0 + \frac{1}{101}y = 1$

$0 + \frac{1}{101}y = 1$ $\displaystyle 0 + \frac{1}{101}y = 1$

$\frac{1}{101}y = 1$ $\displaystyle \frac{1}{101}y = 1$

$\frac{1}{101}y \cdot 101 = 1\cdot 101$ $\displaystyle \frac{1}{101}y \cdot 101 = 1\cdot 101$

y = 101 $\displaystyle y = 101$

This makes (b) the greater quantity

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Example Question #12 : How To Find The Solution To An Equation

$\left \lfloor N\right \rfloor$ $\displaystyle \left \lfloor N\right \rfloor$ refers to the greatest integer less than or equal to $\displaystyle N$.

$\displaystyle x$ and $\displaystyle y$ are integers. Which is greater?

(a) $\left \lfloor x+ y\right \rfloor$ $\displaystyle \left \lfloor x+ y\right \rfloor$

(b) $\left \lfloor x\right \rfloor + \left \lfloor y\right \rfloor$ $\displaystyle \left \lfloor x\right \rfloor + \left \lfloor y\right \rfloor$

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

If $\displaystyle N$ is an integer, then $\left \lfloor N\right \rfloor = N$ $\displaystyle \left \lfloor N\right \rfloor = N$ by definition.

Since x, y $\displaystyle x, y$, and, by closure, x + y $\displaystyle x + y$ are all integers,

$\left \lfloor x+ y\right \rfloor = x + y$ $\displaystyle \left \lfloor x+ y\right \rfloor = x + y$ and $\left \lfloor x\right \rfloor + \left \lfloor y\right \rfloor = x + y$ $\displaystyle \left \lfloor x\right \rfloor + \left \lfloor y\right \rfloor = x + y$, making (a) and (b) equal.

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Example Question #15 : Algebraic Concepts

Consider the line of the equation $\displaystyle 5x + 4y = -200$.

Which is the greater quantity?

(a) The $\displaystyle x$-coordinate of the $\displaystyle x$-intercept.

(b) The $\displaystyle y$-coordinate of the $\displaystyle y$-intercept.

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

(a) To find the $\displaystyle x$-coordinate of the $\displaystyle x$-intercept, substitute y = 0 $\displaystyle y = 0$:

$\displaystyle 5x + 4y = -200$

$5x + 4 \cdot 0 = -200$ $\displaystyle 5x + 4 \cdot 0 = -200$

$\displaystyle 5x + 0 = -200$

$\displaystyle 5x = -200$

$5x\div 5= -200\div 5$ $\displaystyle 5x\div 5= -200\div 5$

x = -40 $\displaystyle x = -40$

(b) To find the $\displaystyle y$-coordinate of the $\displaystyle y$-intercept, substitute x = 0 $\displaystyle x = 0$:

$\displaystyle 5x + 4y = -200$

$5 \cdot 0 + 4y = -200$ $\displaystyle 5 \cdot 0 + 4y = -200$

$\displaystyle 0 + 4y = -200$

$\displaystyle 4y = -200$

$4y \div 4= -200 \div 4$ $\displaystyle 4y \div 4= -200 \div 4$

y = -50 $\displaystyle y = -50$

Therefore (a) is the greater quantity.

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Example Question #691 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

$\displaystyle | 45 - t | = 60$

$\displaystyle | 45 - u | = 50$

Which is the greater quantity?

(a) $\displaystyle t$

(b) $\displaystyle u$

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Each can be rewritten as a compound statement. Solve separately:

$\displaystyle | 45 - t | = 60$

$45 - t =- 60 \textrm{ or }45 - t = 60$ $\displaystyle 45 - t =- 60 \textrm{ or }45 - t = 60$

$\displaystyle 45 - t =- 60$

$\displaystyle 45 - t -45 =- 60 -45$

$\displaystyle -t = -105$

t = 105 $\displaystyle t = 105$

$\displaystyle 45 - t = 60$

$\displaystyle 45 - t -45 =60 -45$

-t = 15 $\displaystyle -t = 15$

t = -15 $\displaystyle t = -15$

Similarly:

$\displaystyle | 45 - u | = 50$

$45 - u = -50 \textrm{ or } 45 - u = 50$ $\displaystyle 45 - u = -50 \textrm{ or } 45 - u = 50$

$\displaystyle 45 - u = -50$

$\displaystyle 45 - u -45 = -50 -45$

$\displaystyle - u = -95$

u =95 $\displaystyle u =95$

$\displaystyle 45 - u = 50$

$\displaystyle 45 - u -45 = 50 -45$

- u = 5 $\displaystyle - u = 5$

u = -5 $\displaystyle u = -5$

Therefore, it cannot be determined with certainty which of $\displaystyle t$ and $\displaystyle u$ is the greater.

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Example Question #13 : How To Find The Solution To An Equation

$\displaystyle | t - 10 | = 40$

Which is the greater quantity?

(a) $\displaystyle t$

(b) $\displaystyle 50$

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

If $\displaystyle | t - 10 | = 40$, then either $\displaystyle t - 10 = 40$ or $\displaystyle t - 10 = -40$. Solve for $\displaystyle t$ in both equations:

$\displaystyle t - 10 = 40$

$\displaystyle t - 10 + 10 = 40 + 10$

t = 50 $\displaystyle t = 50$

$\displaystyle t - 10 = -40$

$\displaystyle t - 10 + 10 = -40 + 10$

t = -30 $\displaystyle t = -30$

Therefore, either (a) and (b) are equal or (b) is the greater quantity, but it cannot be determined with certainty.

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Example Question #12 : How To Find The Solution To An Equation

$\displaystyle t^3 = -125$

Which is the greater quantity?

(a) $\displaystyle t$

(b) $\displaystyle -5$

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

Correct answer:

(a) and (b) are equal

Explanation:

$\displaystyle t^3 = -125$

$\sqrt[3]{t^3 }=\sqrt[3]{ -125}$ $\displaystyle \sqrt[3]{t^3 }=\sqrt[3]{ -125}$

$\sqrt[3]{t^3 }= -\sqrt[3]{ 125}$ $\displaystyle \sqrt[3]{t^3 }= -\sqrt[3]{ 125}$

t= -5 $\displaystyle t= -5$

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Example Question #12 : Algebraic Concepts

Consider the line of the equation $\displaystyle 4x + 5y = 100$.

Which is the greater quantity?

(a) The $\displaystyle x$-coordinate of the $\displaystyle x$-intercept

(b) The $\displaystyle y$-coordinate of the $\displaystyle y$-intercept

Possible Answers:

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

(a) To find the $\displaystyle x$-coordinate of the $\displaystyle x$-intercept, substitute y = 0 $\displaystyle y = 0$:

$\displaystyle 4x + 5y = 100$

$4x + 5 \cdot 0 = 100$ $\displaystyle 4x + 5 \cdot 0 = 100$

$\displaystyle 4x + 0 = 100$

$\displaystyle 4x = 100$

$4x \div 4 = 100 \div 4$ $\displaystyle 4x \div 4 = 100 \div 4$

x = 25 $\displaystyle x = 25$

(b) To find the $\displaystyle y$-coordinate of the $\displaystyle y$-intercept, substitute x = 0 $\displaystyle x = 0$:

$\displaystyle 4x + 5y = 100$

$4\cdot 0 + 5y = 100$ $\displaystyle 4\cdot 0 + 5y = 100$

$\displaystyle 5y = 100$

$5y \div 5 = 100\div 5$ $\displaystyle 5y \div 5 = 100\div 5$

y = 20 $\displaystyle y = 20$

(a) is the greater quantity.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Equations

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #681 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #12 : Equations

Example Question #11 : How To Find The Solution To An Equation

Example Question #12 : Equations

Example Question #12 : How To Find The Solution To An Equation

Example Question #15 : Algebraic Concepts

Example Question #691 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #13 : How To Find The Solution To An Equation

Example Question #12 : How To Find The Solution To An Equation

Example Question #12 : Algebraic Concepts

All ISEE Upper Level Quantitative Resources

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