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

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Example Questions

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Example Question #1 : Word Problems

John and Tom are farmers. John can produce $1500\ lb$ of an organic product in six months. Tom can produce $3500\ lb$ of the organic product in one year. How many pounds of organic product can they produce in three years together? (Assume that they can work and produce the organic product all year round.)

Possible Answers:

$20000\ lb$

$19000\ lb$

$19500\ lb$

$18000\ lb$

$18500\ lb$

Correct answer:

$19500\ lb$

Explanation:

John can produce $1500\ lb$ in six month. Therefore we can set up a proportion:

$\frac{1500 \ lb}{0.5\ yr}=\frac{x \ lb}{3\ yr}\Rightarrow 0.5x=3\times 1500\Rightarrow x=9000\ lb$

Tom can produce $3500\ lb$ in one year. Then we can set up another proportion:

$\frac{3500 \ lb}{1\ yr}=\frac{y \ lb}{3\ yr}\Rightarrow 1\times y=3\times 3500 \Rightarrow y=10500\ lb$

$x+y=9000+10500=19500\ lb$

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Example Question #71 : Equations

Solve for .

$\frac{1}{7}x-7=49$

Possible Answers:

295

290

300

296

294

Correct answer:

294

Explanation:

$\frac{1}{7}x-7+7=49-7\Rightarrow \frac{x}{7}=42\Rightarrow x=42\times 7=294$

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Example Question #71 : Algebraic Concepts

Solve the set of equations:

3x-2y=9

4x+y=1

Possible Answers:

x=-1, y=-3

x=-1, y=3

x=1, y=3

x=-3, y=1

x=1, y=-3

Correct answer:

x=1, y=-3

Explanation:

Solve the second equation for :

y=1-4x

Now substitute this expression into the first equation:

$3x-2y=9\Rightarrow 3x-2(1-4x)=9\Rightarrow 3x-2+8x=9\Rightarrow 11x=11$

$\Rightarrow x=1$

Substitute this new value for into the first equation:

$3(1)-2y=9\Rightarrow 3-9=2y\Rightarrow y=-\frac{6}{2}=-3$

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Example Question #73 : How To Find The Solution To An Equation

3x+4y=25

If x=4 , find .

Possible Answers:

Correct answer:

Explanation:

$3x+4y=25\Rightarrow 3\times 3+4y=25\Rightarrow 9+4y=25\Rightarrow (9-9)+4y=25-9$

$\Rightarrow 4y=16\Rightarrow y=\frac{16}{4}=4$

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Example Question #74 : How To Find The Solution To An Equation

$\frac{1}{3}x+4y^2=45$

If x=15 , find .

Possible Answers:

$\sqrt{10}$

$\sqrt{15}$

$\sqrt{5}$

Correct answer:

$\sqrt{10}$

Explanation:

$\frac{1}{3}x+4y^2=45\Rightarrow \frac{15}{3}+4y^2=45\Rightarrow 5+4y^2=45$

$\Rightarrow (5-5)+4y^2=45-5\Rightarrow 4y^2=40\Rightarrow y^2=\frac{40}{4}=10$

Take the square root of both sides:

$y=\sqrt{10}$

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Example Question #75 : How To Find The Solution To An Equation

Solve the set of equations:

x+7y=39

x-2y=-6

Possible Answers:

x=4, y=-5

x=5, y=4

x=-4, y=-5

x=-4, y=5

x=4, y=5

Correct answer:

x=4, y=5

Explanation:

Solve the second equation for :

$x+6=2y\Rightarrow y=\frac{x+6}{2}$

Now substitute this expression into the first equation:

$x+7y=39\Rightarrow x+7(\frac{x+6}{2})=39\Rightarrow x+\frac{7x}{2}+\frac{7\times 6}{2}=39$

$\Rightarrow \frac{2+7}{2}x+21=39\Rightarrow \frac{9x}{2}=39-21\Rightarrow \frac{9x}{2}=18$

$\Rightarrow x=\frac{18\times 2}{9}=4$

Substitute this new value of into the first equation:

$x+7y=39\Rightarrow 4+7y=39\Rightarrow 4-4+7y=39-4\Rightarrow 7y=35$

$\Rightarrow y=\frac{35}{7}=5$

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Example Question #76 : How To Find The Solution To An Equation

Give the slope of the line of the equation: $\frac{}{}$ 5x -7y = 9

Possible Answers:

$-\frac{7}{5}$

$-\frac{5}{7}$

$\frac{7}{5}$

$\frac{5}{7}$

Correct answer:

$\frac{5}{7}$

Explanation:

Rewrite in the slope-intercept form y = mx + b :

5x -7y = 9

5x-5x -7y = 9-5x

-7y = -5x+9

$-7y \div (-7)=\left ( -5x+9 \right )\div (-7)$

$y=\frac{-5}{-7}\cdot x+ \frac{ 9 }{-7}$

$y=\frac{5}{7} x- \frac{ 9 }{7}$

The slope is the coefficient of , which is $\frac{5}{7}$ .

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Example Question #77 : How To Find The Solution To An Equation

Give the slope of the line of the equation: 5x = 8y + 7

Possible Answers:

$\frac{5}{8}$

$\frac{5}{7}$

$\frac{8}{5}$

$\frac{7}{5}$

$\frac{8}{7}$

Correct answer:

$\frac{5}{8}$

Explanation:

Rewrite in the slope-intercept form y = mx + b :

8y + 7 = 5x

8y + 7 -7 = 5x-7

8y = 5x-7

$8y \div 8 =\left ( 5x-7 \right )\div 8$

$y =\frac{5x}{8} -\frac{7}{8}$

$y =\frac{5}{8}x -\frac{7}{8}$

The slope is the coefficient of , which is $\frac{5}{8}$ .

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Example Question #71 : How To Find The Solution To An Equation

Solve for :

$5^{6x} = 125 ^{2x-1}$

Possible Answers:

x = 1

The equation has no solution.

x = 2

x = 6

x = 3

Correct answer:

The equation has no solution.

Explanation:

$125 = 5^{3}$ , so we can rewrite this equation as:

$5^{6x} = 125 ^{2x-1}$

$5^{6x} = \left ( 5^{3}\right ) ^{2x-1}$

$5^{6x} =5^{3\left ( 2x-1 \right )}$

$5^{6x} =5^{ 6x-3 }$

Therefore, we set the exponents equal.

6x = 6x - 3

6x -6x= 6x-6x - 3

0 = -3

This is identically false. This means that the equation has no solution.

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Example Question #72 : How To Find The Solution To An Equation

Solve for , giving all real solutions:

$x^{3} + 18x = 9x^{2}$

Possible Answers:

$x \in \left \{0,2,9\right \}$

$x \in \left \{ 0,3,6\right \}$

The equation has no solution.

$x \in \left \{ 3,6\right \}$

$x \in \left \{2,9\right \}$

Correct answer:

$x \in \left \{ 0,3,6\right \}$

Explanation:

Write the equation in standard form:

$x^{3} + 18x = 9x^{2}$

$x^{3} - 9x^{2} + 18x = 9x^{2} - 9x^{2}$

$x^{3} - 9x^{2} + 18x = 0$

Factor out the greatest common factor of :

$x \left ( x^{2} - 9x + 18 \right ) = 0$

Factor the trinomial by writing (x+?)(x+?) , replacing the question marks with two integers with product and sum . These integers are -6,-3 , so the above becomes

x( x-6) (x-3)= 0 .

Set each of the three factors equal to and solve separately:

x = 0

$x - 6 = 0 \Rightarrow x = 6$

$x - 3 = 0 \Rightarrow x = 3$

The solution set is $x \in \left \{ 0,3,6\right \}$ .

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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 #1 : Word Problems

Example Question #71 : Equations

Example Question #71 : Algebraic Concepts

Example Question #73 : How To Find The Solution To An Equation

Example Question #74 : How To Find The Solution To An Equation

Example Question #75 : How To Find The Solution To An Equation

Example Question #76 : How To Find The Solution To An Equation

Example Question #77 : How To Find The Solution To An Equation

Example Question #71 : How To Find The Solution To An Equation

Example Question #72 : How To Find The Solution To An Equation

All ISEE Upper Level Math Resources

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