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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

Example Question #121 : How To Find The Solution To An Equation

If is negative and |4y - 27| = 80 , then what is ?

Possible Answers:

y = -428

The equation has no negative solution.

$y = - 26 \frac{3}{4}$

$y = -13 \frac{1}{4}$

y = -212

Correct answer:

$y = -13 \frac{1}{4}$

Explanation:

|4y - 27| = 80 , so either

4y - 27 = - 80 or 4y - 27 = 80

We solve both equations separately:

4y - 27 = - 80

4y - 27 + 27 = - 80 + 27

4y = -53

$\frac{4y}{4} = \frac{-53}{4}$

$y = -13 \frac{1}{4}$

4y - 27 = 80

4y - 27 + 27 = 80 + 27

4y = 107

$\frac{4y}{4} = \frac{107}{4}$

$y = 26 \frac{3}{4}$

Since the negative solution is being requested, we choose $y = -13 \frac{1}{4}$ .

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Example Question #122 : How To Find The Solution To An Equation

Give the solution set of the equation:

|3t + 12| - 15 = 34

Possible Answers:

$\left \{-12\frac{1}{3},12\frac{1}{3} \right \}$

$\left \{-10\frac{1}{3},12\frac{1}{3} \right \}$

$\left \{-20 \frac{1}{3},12\frac{1}{3} \right \}$

$\left \{-10\frac{1}{3},10\frac{1}{3} \right \}$

$\left \{-20 \frac{1}{3},20\frac{1}{3} \right \}$

Correct answer:

$\left \{-20 \frac{1}{3},12\frac{1}{3} \right \}$

Explanation:

|3t + 12| - 15 = 34

|3t + 12| - 15 + 15 = 34 + 15

|3t + 12| =49

Either

3t + 12 = -49 or 3t + 12 = 49 ,

so we solve the equations separately:

3t + 12 = -49

3t + 12 - 12 = -49 - 12

3t = -61

$\frac{3t}{3} =\frac{ -61}{3}$

$t = -20 \frac{1}{3}$

3t + 12 = 49

3t + 12 - 12 = 49 - 12

3t = 37

$\frac{3t}{3} =\frac{37}{3}$

$t = 12\frac{1}{3}$

The solution set is $\left \{-20 \frac{1}{3},12\frac{1}{3} \right \}$

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Example Question #121 : How To Find The Solution To An Equation

|x + 5| = |x | - 5

Which of the folllowing is a true statement?

Possible Answers:

The equation has three solutions.

The equation has no solution.

The equation has infinitely many solutions.

The equation has one solution.

The equation has two solutions.

Correct answer:

The equation has infinitely many solutions.

Explanation:

Since the absolute value of a nonnegative number is the number itself, and the absolute value of a negative number is its (positive) opposite, we have to examine up to three cases: $x \ge 0$ , $-5 \le x < 0$ , and x < -5 .

However, let us examine that third case.

This makes and x+5 negative, so the equation can be rewritten:

|x + 5| = |x | - 5

-(x+5) = -x - 5

-x-5= -x - 5

This statement is identically true. Therefore, all values of less than work, and we have already proved that there are infinitely many solutions. We do not need to go further.

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Example Question #124 : How To Find The Solution To An Equation

t+ 13 = u + 8

u + 4 = v - 7

v- 11 = w - 6

Which of the following is a true statement?

Possible Answers:

t = w -19

t = w -21

t = w -13

t = w -27

t = w -11

Correct answer:

t = w -11

Explanation:

t+ 13 = u + 8 ,

t+ 13- 13 = u + 8 - 13

t = u - 5

u + 4 = v - 7

u + 4 - 4 = v - 7 - 4

u = v-11

v- 11 = w - 6

v- 11 + 11= w - 6 + 11

v = w+5

Using two substitutions:

t = u - 5

t = v-11 - 5 = v-16

t = v-16 = w+5 -16 = w -11

The correct choice is t = w -11 .

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Example Question #125 : How To Find The Solution To An Equation

3t = 7u

4u = 7v

9v = 2w

Which of the following is a true statement?

Possible Answers:

189t = 56w

27t = 8w

6t = w

147t = 8w

54t = 49 w

Correct answer:

54t = 49 w

Explanation:

3t = 7u

$\frac{3t}{3} =\frac{ 7u}{3}$

$t =\frac{ 7}{3}u$

Similarly,

4u = 7v

$u = \frac{7}{4}v$

9v = 2w

$v = \frac{2}{9}w$

By substitution:

$t =\frac{ 7}{3}u =\frac{ 7}{3} \cdot \frac{7}{4}v =\frac{ 7}{3} \cdot \frac{7}{4} \cdot \frac{2}{9}w$

$t = \frac{49}{54}w$

$54 \cdot t =54 \cdot \frac{49}{54}w$

54t = 49 w

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Example Question #126 : How To Find The Solution To An Equation

$8 ^{3x- 8} = 4 ^{y}$

Express in terms of .

Possible Answers:

$y = 2x-\frac{16}{3}$

$y = \frac{3}{2}x- 4$

y = 6x- 16

$y = \frac{9}{2}x- 12$

y = 3x-8

Correct answer:

$y = \frac{9}{2}x- 12$

Explanation:

$8 ^{3x- 8} = 4 ^{y}$

$\left (2^{3} \right )^{3x- 8} = (2^{2}) ^{y}$

$2^{3(3x- 8)} = 2^{2y}$

2y = 3(3x-8)

2y = 9x-24

$\frac{2y }{2}= \frac{9x-24}{2}$

$y = \frac{9}{2}x- 12$

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Example Question #122 : How To Find The Solution To An Equation

$6 ^{x} \cdot 36 ^{y} = 216 ^{t}$

Which of the following is true of ?

Possible Answers:

$t = \frac{1}{3}x+ \frac{2}{3}y$

t = 3x+6y

t = x+2y

$t = \frac{2}{3}x+ \frac{1}{3}y$

t = 6x+ 3y

Correct answer:

$t = \frac{1}{3}x+ \frac{2}{3}y$

Explanation:

$6 ^{x} \cdot 36 ^{y} = 216 ^{t}$

$6 ^{x} \cdot\left ( 6 ^{2} \right )^{y} = \left (6^{3} \right )^{t}$

$6 ^{x} \cdot 6 ^{2 y} = 6^{3 t}$

$6 ^{x + 2 y} = 6^{3 t}$

3t = x+2y

$\frac{3t }{3}= \frac{x+2y}{3}$

$t = \frac{1}{3}x+ \frac{2}{3}y$

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Example Question #123 : How To Find The Solution To An Equation

$\frac{4 ^{x} }{8 ^{y} } = 16^{t}$

Which of the following is true of ?

Possible Answers:

t = 4x-2y

$t = 2x-\frac{4}{3} y$

$t = \frac{1}{2}x- \frac{3}{4}y$

t = 8x-12y

None of the other responses gives a correct answer.

Correct answer:

$t = \frac{1}{2}x- \frac{3}{4}y$

Explanation:

$\frac{4 ^{x} }{8 ^{y} } = 16^{t}$

$\frac{\left (2^{2} \right )^{x} }{\left (2^{3} \right ) ^{y} } =\left ( 2^{4} \right ) ^{t}$

$\frac{ 2^{2x} }{ 2^{3y} } = 2^{4t}$

$2^{2x -3y} = 2^{4t}$

4t = 2x-3y

$\frac{4t }{4}= \frac{2x-3y}{4}$

$t = \frac{1}{2}x- \frac{3}{4}y$

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Example Question #124 : How To Find The Solution To An Equation

Function 4

Define to be the function graphed in the figure above, and g(x) = 4x+8 .

Evaluate $(g \circ f) (-3)$

Possible Answers:

$(g \circ f) (-3) = 16$

is outside the domain of $g \circ f$

$(g \circ f) (-3) = 4$

$(g \circ f) (-3) = -4$

$(g \circ f) (-3) = 8$

Correct answer:

$(g \circ f) (-3) = 16$

Explanation:

$(g \circ f) (-3) = g [f(-3)]$

From the diagram below, it can be seen that f(-3) = 2 .

Function 4a

$(g \circ f) (-3) = g [f(-3)] = g(2)$

g(x) = 4x+8 , so

g(x) = 4 (2)+8 = 8 + 8 = 16

Therefore,

$(g \circ f) (-3) = 16$ .

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Example Question #125 : How To Find The Solution To An Equation

Function 4

Define to be the function graphed in the figure above, and g(x) = 4x+8 .

Evaluate $(f \circ g) (-3)$

Possible Answers:

$(f \circ g) (-3) = 16$

$(f \circ g) (-3) = 4$

$(f \circ g) (-3) = 8$

$(f \circ g) (-3) = -4$

is outside the domain of $f \circ g$

Correct answer:

$(f \circ g) (-3) = 4$

Explanation:

$\left (f \circ g \right ) (-3) = f \left [ g (-3)\right ]$

g(x) = 4x+8

g(-3) = 4 (-3)+8 = -12 + 8 = -4

$f \left [ g (-3)\right ] = f(-4)$ .

Examine the diagram below.

Function 4a

As can be seen, f(-4)= 4 . Therefore, $(f \circ g) (-3) = 4$ .

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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 #121 : How To Find The Solution To An Equation

Example Question #122 : How To Find The Solution To An Equation

Example Question #121 : How To Find The Solution To An Equation

Example Question #124 : How To Find The Solution To An Equation

Example Question #125 : How To Find The Solution To An Equation

Example Question #126 : How To Find The Solution To An Equation

Example Question #122 : How To Find The Solution To An Equation

Example Question #123 : How To Find The Solution To An Equation

Example Question #124 : How To Find The Solution To An Equation

Example Question #125 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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