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ISEE Upper Level Math : How to find the solution to an equation

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #161 : Equations

Give the solution set of the equation

|x + 17| = | 3 - x| .

Possible Answers:

The equation has no solution

$\left \{ -7 , 10 \right \}$

$\left \{ 10 \right \}$

$\left \{ -7 \right \}$

Correct answer:

$\left \{ -7 \right \}$

Explanation:

If |x + 17| = | 3 - x| , then the expressions within the absolute value bars are either equal to each other or the opposites of each other. That is:

x+ 17 = 3- x

x+ 17 + x - 17 = 3- x + x - 17

2 x = -14

x = -7

x+ 17 = - (3- x)

x+ 17 =-3 + x

x+ 17 - x =-3 + x - x

17 =-3

Since this is a false statement, this yields no solution.

Therefore, the solution set comprises one value - $\left \{ -7 \right \}$ .

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Example Question #162 : Equations

One tenth of the reciprocal of is equal to $2 \frac{6}{7}$ . Evaluate .

Possible Answers:

$y = \frac{2}{700}$

$y = \frac{2}{7}$

$y = \frac{7 } {2}$

$y = \frac{7 } {200}$

Correct answer:

$y = \frac{7 } {200}$

Explanation:

"One tenth of the reciprocal of is equal to $2 \frac{6}{7}$ " can be written in algebraic form as

$\frac{1}{10} \cdot \frac{1}{y} = 2 \frac{6}{7}$

Rewriting and solving:

$\frac{1}{10y} = \frac{20}{7}$

$\frac{1}{10y} \cdot \frac{7y } {20} = \frac{20}{7} \cdot \frac{7y } {20}$

$\frac{1}{10} \cdot \frac{7 } {20} =y$

$\frac{7 } {200} =y$

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Example Question #163 : Equations

$\sqrt{t} + 0.2= \frac{7}{9}$

Which of the following is closest to ?

Possible Answers:

0.5

0.2

0.3

0.4

Correct answer:

0.3

Explanation:

$\sqrt{t} + 0.2= \frac{7}{9}$ , so

$\sqrt{t} = \frac{7}{9} - 0.2$

A reasonable estimate for $\frac{7}{9}$ can be made by dividing 7 by 9, carrying out to three decimal digits, as follows:

Division

We can round this to 0.78, and work accordingly:

$\sqrt{t} \approx 0.78- 0.2 = 0.58$

$t = 0.58 ^{2} = 0.58 \cdot 0.58 = 0.3364$

This makes 0.3 the correct choice.

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Example Question #161 : How To Find The Solution To An Equation

Solve for x.

3x+4=-2

Possible Answers:

x=0

x=-1

x=1

x=-2

Correct answer:

x=-2

Explanation:

3x+4=-2

3x+4-4=-2-4

3x=-6

$\frac{3x}{3}=\frac{-6}{3}$

x=-2

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Example Question #165 : Equations

Solve the following equation for w when h is equal to 5

$\frac{h^3-25}{5}=w+2$

Possible Answers:

No real solutions

w=20

w=18

w=-5

Correct answer:

w=18

Explanation:

Solve the following equation for w when h is equal to 5

$\frac{h^3-25}{5}=w+2$

Let's begin by plugging in 5 for h, then just follow the algebra:

$\frac{5^3-25}{5}=w+2$

$\frac{125-25}{5}=w+2$

$\frac{100}{5}=w+2$

20=w+2

So our answer is:

w=18

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Example Question #162 : How To Find The Solution To An Equation

Solve the following equation for u:

12u^2=144

Possible Answers:

u=12

u=4

No real solutions.

$u=\sqrt{12}$

Correct answer:

$u=\sqrt{12}$

Explanation:

Solve the following equation for u:

12u^2=144

Begin by dividing both sides by 12:

u^2=12

Next, take the square root of both sides:

$u=\sqrt{12}$

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Example Question #863 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Define f(x) = 8x - 17 .

If (f +g) (5 ) = 72 , which of the following might be a valid definition of the function g(x) ?

Possible Answers:

$g (x) =( x+ 4 )^{2}$

$g (x) =( x+ 1)^{2}$

$g (x) =( x+ 2 )^{2}$

$g (x) =( x+ 3 )^{2}$

Correct answer:

$g (x) =( x+ 2 )^{2}$

Explanation:

f(x) = 8x - 17 , so, substituting 5 for , we see that

$f(5) = 8 \cdot 5 - 17 = 40 - 17 = 23$

(f +g) (5 ) = f(5)+ g(5) , by definition, so

(f +g) (5 ) = 72

f(5)+ g(5) = 72

23 + g(5) = 72

23 + g(5) - 23 = 72 - 23

g(5) = 49

Examine all four alternatives by, again, substituting 5 for , and finding for which one g(5) = 49 :

$g (x) =( x+ 1)^{2}$

$g (5) =( 5+ 1)^{2} = 6 ^{2} = 36$

$g (x) =( x+2)^{2}$

$g (5) =( 5+ 2)^{2} =7 ^{2} = 49$

$g (x) =( x+3)^{2}$

$g (5) =( 5+ 3)^{2} =8 ^{2} = 64$

$g (x) =( x+4)^{2}$

$g (5) =( 5+4)^{2} =9 ^{2} = 81$

As can be seen, of the four choices, defining $g (x) =( x+2)^{2}$ is the one that results in the correct value.

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Example Question #163 : Equations

Solve for y if z is equal to 5.

625-z^3=2y+400

Possible Answers:

y=50

y=100

y=500

Not enough information to solve.

Correct answer:

y=50

Explanation:

Solve for y if z is equal to 5.

625-z^3=2y+400

Let's begin by plugging in 5 for z. then we can simplify and use algebra to find y:

625-5^3=2y+400

625-125=2y+400

500=2y+400

500-400=2y

100=2y

$\frac{100}{2}=y$

And so we get:

y=50

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Example Question #161 : Equations

Define f(x) = 3x - 7 . The graph of g(x) is a line with -intercept (0, 4 ) .

Which is the definition of g(x) if $(f \circ g) (6) =11$ ?

Possible Answers:

g(x)= 3 x+ 4

g(x)= -3 x+ 4

$g(x)= \frac{1}{3}x+ 4$

$g(x)= - \frac{1}{3}x+ 4$

Correct answer:

$g(x)= \frac{1}{3}x+ 4$

Explanation:

By definition,

$(f \circ g) (6) = f[g(6)]$ .

Let c = g(6) ; then

$(f \circ g) (6) = f(c)$

f(c) = 11

3c - 7 = 11

Solving for :

3c - 7 + 7 = 11 + 7

3c= 18

$3c \div 3 = 18 \div 3$

c = 6

Therefore,

g(6) = 6 .

The graph of g(x) therefore includes the point with coordinates (6,6) as well as that with coordinates (0, 4 ) . The slope of the line can be found by setting $x_{1} = 0, y_{1} = 4 , x_{2} = y_{2} = 6$ in the following slope formula"

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1} }$

$m = \frac{6-4}{6-0 } = \frac{2}{6} = \frac{1}{3}$ .

Substitute this for and 4 for in the slope-intercept form g(x)= mx+ b of the line; the definition of g(x) is $g(x)= \frac{1}{3}x+ 4$ .

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Example Question #163 : How To Find The Solution To An Equation

Solve the following equation for l.

3l+24=57

Possible Answers:

l=11

l=23

l=13

l=12.5

Correct answer:

l=11

Explanation:

Solve the following equation for l.

3l+24=57

We just need a little algebra to rearrange this.

First, subtract 24 from both sides.

3l=57-24=33

3l=33

Next, divide both sides by 3

$l=\frac{33}{3}=11$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : How to find the solution to an equation

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #161 : Equations

Example Question #162 : Equations

Example Question #163 : Equations

Example Question #161 : How To Find The Solution To An Equation

Example Question #165 : Equations

Example Question #162 : How To Find The Solution To An Equation

Example Question #863 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #163 : Equations

Example Question #161 : Equations

Example Question #163 : How To Find The Solution To An Equation

All ISEE Upper Level Math Resources

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