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ACT Math : Linear / Rational / Variable Equations

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Example Questions

Example Question #31 : Linear / Rational / Variable Equations

What is the slope of the line 7y – 4x = 27

Possible Answers:

$-\frac{7}{4}$ $\displaystyle -\frac{7}{4}$

$\frac{4}{7}$ $\displaystyle \frac{4}{7}$

$\frac{7}{4}$ $\displaystyle \frac{7}{4}$

$-\frac{4}{7}$ $\displaystyle -\frac{4}{7}$

Correct answer:

$\frac{4}{7}$ $\displaystyle \frac{4}{7}$

Explanation:

Adding 4x to both sides of the equation and dividing by 7 yields a slope of 4/7.

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Example Question #31 : Equations / Inequalities

If you drove at an average speed of 78 miles per hour, what distance, in miles, did you drive in 140 minutes?

Possible Answers:

182

1.8

4,680

156

Correct answer:

182

Explanation:

140 minutes is 7/3 of an hour. Multiplied by the speed of 78mph, we obtain 182 miles traveled.

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Example Question #32 : Linear / Rational / Variable Equations

3x + 9i² – 5x = 17

What is x?

Possible Answers:

–1

–13

–4

Correct answer:

–13

Explanation:

i = $\sqrt{-1}$ $\displaystyle \sqrt{-1}$

i²= -1

3x + 9i² – 5x = 17

3x + 9(–1) – 5x = 17

–2x – 9 = 17

–2x = 26

x = –13

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Example Question #33 : Linear / Rational / Variable Equations

Solve for x:

3x + 4y = 26

–5x + 12y = 14

Possible Answers:

$-\frac{32}{7}$ $\displaystyle -\frac{32}{7}$

$-\frac{7}{32}$ $\displaystyle -\frac{7}{32}$

$\frac{7}{32}$ $\displaystyle \frac{7}{32}$

$\frac{46}{7}$ $\displaystyle \frac{46}{7}$

$\frac{32}{7}$ $\displaystyle \frac{32}{7}$

Correct answer:

$\frac{32}{7}$ $\displaystyle \frac{32}{7}$

Explanation:

Eliminate y and solve for x.

3x + 4y = 26 (multiply by –3)

–5x + 12y = 14

(–3)3x +(–3) 4y = (–3)26

–5x + 12y = 14

–9x +-12y = –78

–5x + 12y = 14

–14x + 0y = –64

x = –64/–14 = 32/7

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Example Question #34 : Linear / Rational / Variable Equations

Michael is counting his money. He notices he has one more quarter than he does dimes, as well as one less nickel than dimes. The total cash he has is $2.60. How many coins does he have in total?

Possible Answers:

$\displaystyle 24$

$\displaystyle 21$

$\displaystyle 18$

$\displaystyle 15$

$\displaystyle 27$

Correct answer:

$\displaystyle 18$

Explanation:

The general form for money problems is $V_{1}N_{1} + V_{2}N_{2} + V_{3}N_{3} =$ $\displaystyle V_{1}N_{1} + V_{2}N_{2} + V_{3}N_{3} =$ $\$\ total$ $\displaystyle \$\ total$, where $\displaystyle V$ is the value of the coin and $\displaystyle N$ is the number of coins.

Let $\displaystyle x$ = # of dimes, x + 1 $\displaystyle x + 1$ = # of quarters, and x - 1 $\displaystyle x - 1$ = # of nickels.

So, $\displaystyle 0.10x + 0.25(x + 1) + 0.05(x - 1) = 2.60$, and solving gives x = 6 $\displaystyle x = 6$. Therefore there are 6 dimes, 7 quarters, and 5 nickels, giving 18 coins in total.

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Example Question #35 : Linear / Rational / Variable Equations

What value of $\displaystyle z$ will satisfy the equation

$\displaystyle 5(z+3)= 2z+12$

Possible Answers:

3/7 $\displaystyle 3/7$

$\displaystyle 12$

-3/7 $\displaystyle -3/7$

$\displaystyle -1$

$\displaystyle 1$

Correct answer:

$\displaystyle -1$

Explanation:

The answer is $\displaystyle -1$.

The solve this equation, first distribute the $\displaystyle 5$ to obtain $\displaystyle 5z+15= 2z+12$

Proceed to subtract $\displaystyle 2z$ from both sides to get $\displaystyle 3z+15=12$

Subtract $\displaystyle 15$ from both sides to leave 3z=-3 $\displaystyle 3z=-3$

Divide both sides by $\displaystyle 3$ to get the answer, z=-1 $\displaystyle z=-1$

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Example Question #36 : Linear / Rational / Variable Equations

Given that $\displaystyle 2x+1=3x-2$, what is the value of $\displaystyle x$ ?

Possible Answers:

$\displaystyle 2$

$\displaystyle -1$

$\displaystyle 3$

$\frac{5}{7}$ $\displaystyle \frac{5}{7}$

$\frac{3}{2}$ $\displaystyle \frac{3}{2}$

Correct answer:

$\displaystyle 3$

Explanation:

First, we must solve the equation for $\displaystyle x$ by subtracting $\displaystyle 2x$ from both sides:

$\displaystyle 2x-2x+1=3x-2x-2$

1=x-2 $\displaystyle 1=x-2$

Then we must add $\displaystyle 2$ to both sides:

$\displaystyle 1+2=x-2+2$

3=x $\displaystyle 3=x$

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

Internet service costs $0.50 per minute for the first ten minutes and is $0.20 a minute thereafter. What is the equation that represents the cost of internet in dollars when time is greater than 10 minutes?

Possible Answers:

5.00 $\displaystyle 5.00$

$\displaystyle 5.00 + 0.20 (x-10)$

$\displaystyle 10 + 0.20 (x-10)$

$\displaystyle 5.00 + 0.20 (x)$

$\displaystyle 5 + 0.20 (x+10)$

Correct answer:

$\displaystyle 5.00 + 0.20 (x-10)$

Explanation:

The first ten minutes will cost $5. From there we need to apply a $0.20 per-minute charge for every minute after ten. This gives

$\$0.20(x-10)+5$ $\displaystyle \$0.20(x-10)+5$.

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Example Question #92 : Equations / Inequalities

John goes on a trip of $\displaystyle b$ kilometers at a speed of $\displaystyle c$ kilometers an hour. How long did the trip take?

Possible Answers:

c/b $\displaystyle c/b$

b/c $\displaystyle b/c$

c+b $\displaystyle c+b$

c-b $\displaystyle c-b$

b-c $\displaystyle b-c$

Correct answer:

b/c $\displaystyle b/c$

Explanation:

If we take the units and look at division, $\displaystyle miles/(miles/hour)$ will yield hours as a unit. Therefore the answer is b/c $\displaystyle b/c$.

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Example Question #1809 : Sat Mathematics

With a $25\ mph$ $\displaystyle 25\ mph$ head wind a plane can fly a certain distance in five hours. The return flight takes an hour less. How fast was the plane flying?

Possible Answers:

$175\ mph$ $\displaystyle 175\ mph$

$225\ mph$ $\displaystyle 225\ mph$

$300\ mph$ $\displaystyle 300\ mph$

$125\ mph$ $\displaystyle 125\ mph$

$275\ mph$ $\displaystyle 275\ mph$

Correct answer:

$225\ mph$ $\displaystyle 225\ mph$

Explanation:

In general, $distance=rate\times time$ $\displaystyle distance=rate\times time$.

The distance is the same going and coming; however, the head wind affects the rate. The equation thus becomes $(r-25)\times 5=(r+25)\times 4$ $\displaystyle (r-25)\times 5=(r+25)\times 4$.

Solving for $r$ $\displaystyle r$ gives $r=225\ mph$ $\displaystyle r=225\ mph$.

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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 #31 : Linear / Rational / Variable Equations

Example Question #31 : Equations / Inequalities

Example Question #32 : Linear / Rational / Variable Equations

Example Question #33 : Linear / Rational / Variable Equations

Example Question #34 : Linear / Rational / Variable Equations

Example Question #35 : Linear / Rational / Variable Equations

Example Question #36 : Linear / Rational / Variable Equations

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

Example Question #92 : Equations / Inequalities

Example Question #1809 : Sat Mathematics

All ACT Math Resources

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