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ISEE Upper Level Quantitative : Equations

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

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

x(2x-9)= 35

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

x(2x-9)= 35

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

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

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

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

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

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

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

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

Set each expression equal to 0 and solve:

2x+5 = 0

2x=-5

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

x-7 = 0

x=7

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

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

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

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

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

0.9x + 0.7y = 100

$0.9x + 0.7 \cdot 0 = 100$

0.9x + 0 = 100

0.9x = 100

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

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

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

0.9x + 0.7y = 100

$0.9 \cdot 0 + 0.7y = 100$

0 + 0.7y = 100

0.7y = 100

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

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

(b) is the greater quantity.

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

| 50 - t | = 30

Which is the greater quantity?

(a)

(b) 0

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

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

50 - t = 30 or 50 - t = -30

Solve both:

50 - t = 30

50 - t - 50 = 30 - 50

-t = -20

t = 20

50 - t = -30

50 - t -50 = -30-50

-t = -80

t = 80

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

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Example Question #14 : How To Find The Solution To An Equation

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

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

Possible Answers:

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

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

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

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

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

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

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

x= 99

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

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

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

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

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

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

y = 101

This makes (b) the greater quantity

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Example Question #15 : How To Find The Solution To An Equation

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

and are integers. Which is greater?

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

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

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

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

Since x, y , and, by closure, x + y are all integers,

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

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Example Question #16 : How To Find The Solution To An Equation

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

Which is the greater quantity?

(a) The -coordinate of the -intercept.

(b) The -coordinate of the -intercept.

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

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

5x + 4y = -200

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

5x + 0 = -200

5x = -200

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

x = -40

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

5x + 4y = -200

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

0 + 4y = -200

4y = -200

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

y = -50

Therefore (a) is the greater quantity.

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Example Question #17 : How To Find The Solution To An Equation

| 45 - t | = 60

| 45 - u | = 50

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

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

| 45 - t | = 60

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

45 - t =- 60

45 - t -45 =- 60 -45

-t = -105

t = 105

45 - t = 60

45 - t -45 =60 -45

-t = 15

t = -15

Similarly:

| 45 - u | = 50

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

45 - u = -50

45 - u -45 = -50 -45

- u = -95

u =95

45 - u = 50

45 - u -45 = 50 -45

- u = 5

u = -5

Therefore, it cannot be determined with certainty which of and is the greater.

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

| t - 10 | = 40

Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

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

t - 10 = 40

t - 10 + 10 = 40 + 10

t = 50

t - 10 = -40

t - 10 + 10 = -40 + 10

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 #11 : Algebraic Concepts

t^3 = -125

Which is the greater quantity?

(a)

(b)

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) and (b) are equal

Explanation:

t^3 = -125

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

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

t= -5

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Example Question #20 : How To Find The Solution To An Equation

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

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -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 -coordinate of the -intercept, substitute y = 0 :

4x + 5y = 100

$4x + 5 \cdot 0 = 100$

4x + 0 = 100

4x = 100

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

x = 25

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

4x + 5y = 100

$4\cdot 0 + 5y = 100$

5y = 100

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

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 #11 : How To Find The Solution To An Equation

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

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

Example Question #14 : How To Find The Solution To An Equation

Example Question #15 : How To Find The Solution To An Equation

Example Question #16 : How To Find The Solution To An Equation

Example Question #17 : How To Find The Solution To An Equation

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

Example Question #11 : Algebraic Concepts

Example Question #20 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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