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ACT Math : Linear / Rational / Variable Equations

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Example Questions

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Example Question #3 : How To Find The Solution To A Rational Equation With Lcd

In the equation below, , , and are non-zero numbers. What is the value of in terms of and ?

$\frac{1}{m^3}-\frac{1}{k^2}=\frac{1}{p}$

Possible Answers:

$m=(\frac{p+k^2}{2})^3$

$m=(\frac{p+k^2}{pk^2})^{\frac{1}{3}}$

$m=\frac{p^2k^3}{p+k^2}$

$m=(\frac{pk^2}{p+k^2})^{\frac{1}{3}}$

$m=p^{\frac{1}{2}}-k^{\frac{1}{3}}$

Correct answer:

$m=(\frac{pk^2}{p+k^2})^{\frac{1}{3}}$

Explanation:

Pkm_7-21-13

Pkm2_7-21-13

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Example Question #1 : How To Find Out When An Equation Has No Solution

Given the following system, find the solution:

x = y – 2

2x – 2y = 2

Possible Answers:

(1, 1)

(1, 2)

no solution

(0, 1)

(0, 0)

Correct answer:

no solution

Explanation:

When 2 equations in a system have the same slopes, they will either have no solution or infinite solutions. Since the y-intercepts are not the same, there is no solution to this system.

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Example Question #2 : How To Find Out When An Equation Has No Solution

Solve for :

27x-32=-(-27x+27)

Possible Answers:

Infinite Solutions

1.1

No solution

$-9.25\cdot 10^{-2}$

Correct answer:

No solution

Explanation:

27x-32=-(-27x+27)

Like other "solve for x" problems, to begin it, the goal is to get x by itself on one side of the equals sign. In this problem, before doing so, the imaginary -1 in front of (-27x+27) must be distributed.

27x-32=27x-27

Once this is done, you may start to try to get x by itself.

${\color{Red} 27x}-32={\color{Red} 27x}-27$

However, when subtracting 27x from either side and doing the same on the other, the 27x term cancels out. As a result, the equation becomes:

-32=-27

We know this is an untrue statement because these numbers are 5 spaces away from each other on the number line. The final answer is No Solution.

$-32\neq-27$

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Example Question #1 : How To Find Out When An Equation Has No Solution

Given the following system, find the solution:

y = 2x + 3

3y - 5 = 6x

Possible Answers:

$\left (0, \frac{5}{3} \right )$

$\left (- \frac{5}{6}, 0 \right )$

(0, 3)

No solution

$\left (- \frac{2}{3}, 0 \right )$

Correct answer:

No solution

Explanation:

When two equations have the same slope, they will have either no solution or infinite solutions. By putting both equations into the form y = mx +b , we get:

y = 2x + 3

and

$y = 2x +\frac{5}{3}$

With the equations in this form, we can see that they have the same slope, but different y-intercepts. Therefore, there is no solution to this system.

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Example Question #4 : How To Find Out When An Equation Has No Solution

Solve the following equation for :

$2\left ( x - 4\right ) = \frac{4x + 5}{2}$

Possible Answers:

$-\frac{3}{2}$

Infinite solutions

$\frac{13}{2}$

No solution

Correct answer:

No solution

Explanation:

In order to solve for , we must get by itself on one side of the equation.

$2\left ( x - 4\right ) = \frac{4x + 5}{2}$

First, we can distribute the on the left side of the equal sign and the $\frac{1}{2}$ on the right side.

$2x - 8= 2x +\frac{ 5}{2}$

When we try to get by itself, the terms on each side of the equation cancel out, leaving us with:

$- 8= \frac{ 5}{2}$

We know this is an untrue statement, so there is no solution to this equation.

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Example Question #3 : How To Find Out When An Equation Has No Solution

Find the solution to the following equation if x = 3:

y = (4x² - 2)/(9 - x²)

Possible Answers:

no possible solution

Correct answer:

no possible solution

Explanation:

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

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Example Question #5 : How To Find Out When An Equation Has No Solution

Nosol1

Possible Answers:

–1/2

–3

There is no solution

Correct answer:

There is no solution

Explanation:

Nosol2

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Example Question #6 : How To Find Out When An Equation Has No Solution

$\small h\left ( x\right )=\frac{28}{x+4}$

$\small \textup{For which of the following values of}\,x\,\textup{is the above function undefined?}$

Possible Answers:

None of the other answers

$\small 4$

$\small 0$

$\small 28$

$\small -4$

Correct answer:

$\small -4$

Explanation:

A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.

$\small x+4=0$

$\small x=-4$

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Example Question #131 : Gre Quantitative Reasoning

$\frac{x+2}{3}=\frac{x}{3}$ Solve for .

Possible Answers:

No solutions.

Correct answer:

No solutions.

Explanation:

Cross multiplying leaves 3x+6=3x , which is not possible.

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Example Question #1 : How To Find Out When An Equation Has No Solution

Undefined_denom3

I. x = 0

II. x = –1

III. x = 1

Possible Answers:

I only

II and III only

I, II, and III

III only

II only

Correct answer:

I only

Explanation:

Undefined_denom2

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ACT Math : Linear / Rational / Variable Equations

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #3 : How To Find The Solution To A Rational Equation With Lcd

Example Question #1 : How To Find Out When An Equation Has No Solution

Example Question #2 : How To Find Out When An Equation Has No Solution

Example Question #1 : How To Find Out When An Equation Has No Solution

Example Question #4 : How To Find Out When An Equation Has No Solution

Example Question #3 : How To Find Out When An Equation Has No Solution

Example Question #5 : How To Find Out When An Equation Has No Solution

Example Question #6 : How To Find Out When An Equation Has No Solution

Example Question #131 : Gre Quantitative Reasoning

Example Question #1 : How To Find Out When An Equation Has No Solution

All ACT Math Resources

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