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ACT Math : Linear / Rational / Variable Equations

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Example Questions

Example Question #84 : Algebra

Translate into an algebraic expression:

Two fifths of the difference of five times a number and two equals six times the number.

Possible Answers:

$\frac{2}{5}(2x+5)=6x$

$\frac{2}{5}(5x-2)=6x$

$\frac{2}{5}(2x-5)=6x$

$\frac{2}{5}(2-5x)=6x$

$\frac{2}{5}(5x+2)=6x$

Correct answer:

$\frac{2}{5}(5x-2)=6x$

Explanation:

In order to translate the description into an algebraic expression, you must first form the difference (5x-2) , multiply it by $\frac{2}{5}$ , and then set the equation equal to , as follows:

$\frac{2}{5}(5x-2)=6x$

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

A car averages miles per gallon. If gas costs $\$ 3.04$ per gallon, which of the following is closest to how much the gas would cost for this car to travel $1\textup{,}642$ typical miles?

Possible Answers:

$\$51$

$\$155$

$\$156$

$\$50$

$\$17$

Correct answer:

$\$156$

Explanation:

To solve this problem you should divide the total number of miles $\left ( 1\textup{,}642\right )$ by the miles per gallon $\left ( 32\right )$ , and then multiply by the cost of gas per gallon $\left ( \$3.04\right )$ . This gives you $\$155.99$ , which is closest to $\$156$ .

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Example Question #121 : Equations / Inequalities

When $\textup{w}= 6\textup{ and } \textup{z} = 4$ , by how much does the value of $3\textup{w}^{2}-2\textup{z}$ exceed the value of $2\textup{w}^{2} - 3\textup{z}$ ?

Possible Answers:

Correct answer:

Explanation:

You must plug in for $\textup{w}$ and for $\textup{z}$ , this means $3\textup{w}^{2}-2\textup{z}=$ $3\left ( 36\right )-2\left ( 4\right )=100$ and $2\textup{w}^{2}-3\textup{z}=2\left ( 36\right )-3\left ( 4\right )=60$

100-60=40

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

What is the value of $\textup{t}$ when $3\textup{t}+5=7\textup{t}-2$ ?

Possible Answers:

$\frac{7}{10}$

$\frac{4}{7}$

$\frac{7}{4}$

$\frac{4}{3}$

$\frac{3}{4}$

Correct answer:

$\frac{7}{4}$

Explanation:

The first step is to subtract $\textup{3t}$ from both sides so you're left with $5=4\textup{t}-2$ . Next, you add to both sides, giving you $7=4\textup{t}$ . The final step is to divide by , giving you $\frac{7}{4}=\textup{t}$ .

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

Karen has $\$7$ more than times as many dollars as Susan. Karen also has $\frac{1}{2}$ as many dollars as John would if he had more dollars. How many dollars does Susan have, relative to John?

Possible Answers:

$\frac{J}{4}$

$\frac{J-12}{6}$

3J -3

J + 2

6J -4

Correct answer:

$\frac{J-12}{6}$

Explanation:

First, set up our two equations. If Karen has seven dollars more than three times as many dollars as Susan, then K = 3S +7 . If Karen has half as many dollars as John would if he had two more dollars, then $K = \frac{J+2}{2}$ . The Substitution Property of Equality thus says that $3S +7 = \frac{J+2}{2}$ . Now, we simply solve for in terms of .

$3S +7 = \frac{J+2}{2}$

6S +14 = J+2

6S = J-12

$S = \frac{J-12}{6}$

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

If the third day of the year is a Wednesday, what is the 165th day?

Possible Answers:

Thursday

Monday

Wednesday

Friday

Tuesday

Correct answer:

Thursday

Explanation:

If the third day is a Wednesday, then the first day is a Monday. 23 weeks will complete a cycle with the 161st day. Therefore, the fourth day past that, 165, is a Thursday.

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Example Question #1 : How To Find The Solution To A Rational Equation With Lcd

John, Jill, and Jack are splitting a pizza. John eats 1/2 of the pizza and Jill eats 1/3 of the pizza. How much of the pizza is left for Jack?

Possible Answers:

3/5

1/6

1/5

1/3

None

Correct answer:

1/6

Explanation:

We can write an equation for the amount of pizza eaten, with as the amount left for Jack.

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

To solve this equation, we must find the lowest common denominator of 1/2 and 1/3 . We can list the multiples of and to find the least common multiple:

2: 2, 4, 6, 8, 10, 12, ...

3: 3, 6, 12, 15, 18, 21, ...

We can see that the least common multiple of and is , so we can rewrite each of the fractions with a denominator of .

$\frac{1}{2}\cdot \frac{3}{3} = \frac{3}{6}$

$\frac{1}{3}\cdot \frac{2}{2} = \frac{2}{6}$

When we put these fractions back into the equation, we can solve for :

$\frac{3}{6} + \frac{2}{6} + x = 1$

$\frac{5}{6} + x = 1$

$x = \frac{1}{6}$

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Example Question #1 : How To Find The Solution To A Rational Equation With Lcd

What is the slope of a line represented by the equation:

15y + 3x = 4

Possible Answers:

$\frac{4}{3}$

$\frac{3}{4}$

$-\frac{1}{5}$

$\frac{1}{5}$

Correct answer:

$-\frac{1}{5}$

Explanation:

To solve problems where you need to find the slope of a line in a given equation, change the equation so that it matches y-intercept form:

y=mx+b

For this equation, first move the 3x over to the other side of the equation.

15y + 3x = 4

-3x -3x

The equation should now look like this:

15y = -3x + 4

Then, divide by 15 to isolate the variable .

$\frac{15y}{15} =-\frac{3x}{15} + \frac{4}{15} \rightarrow y =-\frac{3}{15}x + \frac{4}{15}$

Then simplify

$y =-\frac{3}{15}x + \frac{4}{15} \rightarrow y =-\frac{1}{5}x + \frac{4}{15}$

Whatever number is before the x in the equation (m) is your slope.

$-\frac{1}{5}$

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Example Question #102 : Equations / Inequalities

Rational_5

Possible Answers:

–1

–2

Correct answer:

Explanation:

Rational_2

Rational_3

Rational_4

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Example Question #51 : Algebra

Linesmb1

Possible Answers:

–b/(m²– 1)

–b/(m + 1)

bm/(m²+ 1)

b/(m²+ 1)

–bm/(m²+ 1)

Correct answer:

b/(m²+ 1)

Explanation:

Linesmb5

Linesmb4

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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 #84 : Algebra

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

Example Question #121 : Equations / Inequalities

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

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

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

Example Question #1 : How To Find The Solution To A Rational Equation With Lcd

Example Question #1 : How To Find The Solution To A Rational Equation With Lcd

Example Question #102 : Equations / Inequalities

Example Question #51 : Algebra

All ACT Math Resources

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