Solving Linear Equations Practice Questions — ACT

What is the value of $x$ that satisfies the equation $\frac{3}{4}(x - 8) = \frac{1}{2}x + 2$ ?

Show answer & explanation

Answer: 32

Answer

The correct value of

x

is

32

.

Distributing

\frac{3}{4}

on the left side yields

\frac{3}{4}x - 6

. Subtracting

\frac{1}{2}x

(which is

\frac{2}{4}x

) from both sides gives

\frac{1}{4}x - 6 = 2

. Adding 6 to both sides gives

\frac{1}{4}x = 8

, and multiplying by 4 yields the correct value of

32

.

Step-by-Step Solution

1

Distribute the fraction

\frac{3}{4}

to both terms inside the parentheses.

\frac{3}{4}x - 6 = \frac{1}{2}x + 2

To simplify the equation by removing the parentheses.

2

Subtract

\frac{1}{2}x

from both sides of the equation.

\frac{1}{4}x - 6 = 2

To collect all terms with the variable

x

on one side of the equation.

3

Add 6 to both sides of the equation.

\frac{1}{4}x = 8

To isolate the term containing the variable.

4

Multiply both sides of the equation by 4.

x = 32

To solve for

x

by eliminating the coefficient of

\frac{1}{4}

.

Key Concept

Solving linear equations with variables on both sides and fractional coefficients

Alternative Method

Multiply the entire equation by the least common denominator of the fractions, which is 4, to clear the fractions before solving:

4 \cdot [\frac{3}{4}(x - 8)] = 4 \cdot [\frac{1}{2}x + 2] \implies 3(x - 8) = 2x + 8

. Then distribute:

3x - 24 = 2x + 8

. Subtract

2x

from both sides:

x - 24 = 8

. Add 24 to both sides:

x = 32

.

Estimated Time:45s

Question 2Question →

A scientist is mixing two solutions in a laboratory. Solution A contains $\frac{1}{4}$ active ingredient by volume, and Solution B contains $\frac{3}{4}$ active ingredient by volume. The scientist needs to mix these to create a $20$ -liter solution containing exactly $\frac{11}{20}$ active ingredient by volume.

Let

x

represent the volume, in liters, of Solution A used. The correct relationship is represented by the linear equation:

\frac{1}{4}x + \frac{3}{4}(20 - x) = 11

An assistant translates the problem description incorrectly, writing the equation as:

\frac{1}{4}x + \frac{3}{4}(x - 20) = 11

What is the absolute difference, in liters, between the value of $x$ obtained from the assistant's incorrect equation and the value of $x$ obtained from the correct equation?

8

30

18

26

44

Show answer & explanation

Answer: 18

Answer

The correct answer is 18, which is the absolute difference between the two solutions.

The correct answer is 18. Solving the correct equation

\frac{1}{4}x + \frac{3}{4}(20 - x) = 11

by multiplying by 4 gives

x + 60 - 3x = 44

, which simplifies to

-2x = -16

and yields

x = 8

. Solving the assistant's incorrect equation

\frac{1}{4}x + \frac{3}{4}(x - 20) = 11

in a similar manner gives

x + 3x - 60 = 44

, which simplifies to

4x = 104

and yields

x = 26

. The absolute difference between these two values is

|26 - 8| = 18

.

Step-by-Step Solution

1

Solve the correct equation for

x

.

Multiply the entire equation by 4 to clear the denominators:

x + 3(20 - x) = 44

. Distribute the 3:

x + 60 - 3x = 44

. Combine like terms:

-2x + 60 = 44

. Subtract 60 from both sides:

-2x = -16

. Divide by -2:

x = 8

.

This determines the correct volume of Solution A needed.

2

Solve the assistant's incorrect equation for

x

.

Multiply the entire equation by 4 to clear the denominators:

x + 3(x - 20) = 44

. Distribute the 3:

x + 3x - 60 = 44

. Combine like terms:

4x - 60 = 44

. Add 60 to both sides:

4x = 104

. Divide by 4:

x = 26

.

This determines the incorrect volume of Solution A obtained by the assistant.

3

Calculate the absolute difference between the two solutions.

|26 - 8| = 18

.

The question asks for the absolute difference between the two values of

x

.

Key Concept

Solving linear equations involving fractions, parentheses, and algebraic manipulation.

Question 3Question →

For what value of $a$ does the equation $\frac{3}{4}(x - 2) - \frac{1}{3}(2x + a) = \frac{1}{12}x - 5$ have infinitely many solutions for $x$ ?

Show answer & explanation

Answer: 10.5

Answer

10.5

Expanding the left side of the equation yields

\frac{3}{4}x - \frac{3}{2} - \frac{2}{3}x - \frac{a}{3}

. Combining the coefficients of

x

gives

(\frac{3}{4} - \frac{2}{3})x = \frac{1}{12}x

. The equation becomes

\frac{1}{12}x - (\frac{3}{2} + \frac{a}{3}) = \frac{1}{12}x - 5

. For a linear equation to have infinitely many solutions, the variable coefficients must be equal and the constant terms must also be equal. Therefore, we equate the constants:

-\frac{3}{2} - \frac{a}{3} = -5

. Multiplying all terms by

-6

to clear the denominators yields

9 + 2a = 30

, which simplifies to

2a = 21

, or

a = 10.5

.

Step-by-Step Solution

1

Expand and simplify the left side of the equation by distributing the fraction coefficients.

\frac{1}{12}x - \frac{3}{2} - \frac{a}{3}

Distributing

\frac{3}{4}

and

-\frac{1}{3}

across their respective parentheses and combining the

x

terms allows us to compare the coefficients on both sides.

2

Equate the constant terms from both sides of the equation.

-\frac{3}{2} - \frac{a}{3} = -5

A linear equation of the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

. Since both

A

and

C

are

\frac{1}{12}

, we set the constant terms equal.

3

Isolate the variable

a

and solve.

a = 10.5

Adding

\frac{3}{2}

to both sides gives

-\frac{a}{3} = -3.5

. Multiplying both sides by

-3

yields the final value.

Key Concept

Solving linear equations with infinitely many solutions by equating coefficients and constant terms.

Estimated Time:2m 0s

Question 4Question →

What is the value of $x$ that satisfies the equation $\frac{2}{3}x - \frac{1}{4} = \frac{5}{12}$ ?

\frac{1}{4}

\frac{9}{16}

\frac{4}{9}

1

12

Show answer & explanation

Answer: $1$

Answer

1

The correct answer is 1. Adding

\frac{1}{4}

to both sides of the equation

\frac{2}{3}x - \frac{1}{4} = \frac{5}{12}

gives

\frac{2}{3}x = \frac{5}{12} + \frac{3}{12} = \frac{8}{12}

. Simplifying

\frac{8}{12}

yields

\frac{2}{3}

, so

\frac{2}{3}x = \frac{2}{3}

. Multiplying both sides by

\frac{3}{2}

isolates

x

, giving

x = 1

.

Step-by-Step Solution

1

Add

\frac{1}{4}

to both sides of the equation to isolate the variable term.

\frac{2}{3}x = \frac{5}{12} + \frac{1}{4}

To solve for

x

, terms containing

x

must be isolated on one side of the equation.

2

Find a common denominator to add the fractions on the right side.

\frac{2}{3}x = \frac{5}{12} + \frac{3}{12} = \frac{8}{12} = \frac{2}{3}

Fractions must have the same denominator to be added.

3

Multiply both sides of the equation by the reciprocal of the coefficient of

x

, which is

\frac{3}{2}

.

x = \frac{2}{3} \cdot \frac{3}{2} = 1

Multiplying a coefficient by its reciprocal yields

1

, isolating the variable

x

.

Key Concept

Solving single-variable linear equations involving fractions by isolating the variable using inverse operations.

Question 5Question →

For what value of the constant $a$ does the linear equation $\frac{a(x - 2)}{3} - \frac{3x - 1}{2} = -\frac{5x + 7}{6}$ have no real solution for $x$ ?

Show answer & explanation

Answer: 2

Answer

The constant

a

must be equal to

2

for the equation to have no solution.

A linear equation in the form

Ax + B = Cx + D

has no solution if the coefficients of the variable are equal (

A = C

) but the constant terms are different (

B \neq D

). Multiplying the given equation by the least common denominator, 6, clears the fractions and yields

2a(x - 2) - 3(3x - 1) = -(5x + 7)

. Expanding both sides results in

2ax - 4a - 9x + 3 = -5x - 7

, which simplifies to

(2a - 9)x + (3 - 4a) = -5x - 7

. Equating the coefficients of

x

gives

2a - 9 = -5

, which solves to

a = 2

. Substituting

a = 2

back into the constants yields a left-side constant of

-5

and a right-side constant of

-7

. Since

-5 \neq -7

, the variable terms cancel out while leaving an inequality, meaning the equation has no solution when

a = 2

.

Step-by-Step Solution

1

Clear the denominators by multiplying the entire equation by the least common denominator, which is 6.

2a(x - 2) - 3(3x - 1) = -(5x + 7)

Multiplying by the LCD simplifies the rational expressions into polynomial terms.

2

Distribute and expand the terms on both sides of the equation.

2ax - 4a - 9x + 3 = -5x - 7

Expanding the terms allows us to group variable terms and constant terms together.

3

Group the

x

terms and constant terms on the left side.

(2a - 9)x + (3 - 4a) = -5x - 7

Structuring the equation in the standard form

Ax + B = Cx + D

makes it easier to compare coefficients.

4

Set the coefficients of

x

on both sides equal to each other.

2a - 9 = -5

For a linear equation to have no solution, the variable terms must cancel out, meaning their coefficients must be identical.

5

Solve for the parameter

a

and verify the constant terms are unequal.

2a = 4 \implies a = 2

. Constant check:

3 - 4(2) = -5

, and

-5 \neq -7

.

If the constant terms were equal, the equation would have infinitely many solutions instead of no solution.

Key Concept

Identifying the parameter value that results in a linear equation having no solution by equating variable coefficients and ensuring constant terms are unequal.

Question 6Question →

For a real number $x$ , the equation $2(x + 5) = 16$ is true. What is the value of the expression $3x - 1$ ?

Show answer & explanation

Answer: 8

Answer

The value of the expression

3x - 1

is 8.

Solving the equation

2(x + 5) = 16

gives

x = 3

. Evaluating the expression

3x - 1

for

x = 3

yields

3(3) - 1 = 8

.

Step-by-Step Solution

1

Distribute the 2 on the left side of the equation.

2x + 10 = 16

To eliminate the parentheses using the distributive property.

2

Subtract 10 from both sides of the equation.

2x = 6

To isolate the variable term on one side of the equation.

3

Divide both sides of the equation by 2.

x = 3

To find the value of

x

.

4

Substitute the value of

x

into the expression

3x - 1

.

3(3) - 1 = 8

To evaluate the final expression as requested by the question.

Key Concept

Solving multi-step linear equations and evaluating algebraic expressions.

Estimated Time:45s

Question 7Question →

An electronics retailer sells tablet computers. The retailer's weekly revenue from tablets, in thousands of dollars, is modeled by the expression $\frac{2}{5}(3x - 4)$ , where $x$ represents the average number of tablets sold per day. The weekly operating costs, in thousands of dollars, are modeled by the expression $\frac{1}{3}\left(2x - \frac{1}{2}\right)$ . The retailer's weekly profit is equal to the profit of a competitor, which is modeled by $\frac{1}{4}(x + 5) - \frac{11}{6}$ thousand dollars. If the retailer and the competitor earn the same weekly profit, what is the value of $12x - 5$ ?

31

3

\frac{767}{17}

51

\frac{367}{97}

Show answer & explanation

Answer: 31

Answer

31

The correct answer is 31. By equating the profit models, we get

\frac{2}{5}(3x - 4) - \frac{1}{3}(2x - 0.5) = \frac{1}{4}(x + 5) - \frac{11}{6}

. Multiplying the entire equation by the least common multiple of the denominators (60) yields

24(3x - 4) - 20(2x - 0.5) = 15(x + 5) - 110

. Expanding the terms gives

72x - 96 - 40x + 10 = 15x + 75 - 110

, which simplifies to

32x - 86 = 15x - 35

. Solving for

x

gives

17x = 51

, or

x = 3

. Evaluating the required expression

12x - 5

for

x = 3

gives

12(3) - 5 = 31

.

Step-by-Step Solution

1

Set up the equation equating the retailer's profit (revenue minus cost) to the competitor's profit.

\frac{2}{5}(3x - 4) - \frac{1}{3}\left(2x - \frac{1}{2}\right) = \frac{1}{4}(x + 5) - \frac{11}{6}

Profit is calculated as revenue minus operating costs. Since the retailer and the competitor earn the same profit, their profit models are equal.

2

Multiply the entire equation by 60 to eliminate all fractional denominators.

24(3x - 4) - 20\left(2x - \frac{1}{2}\right) = 15(x + 5) - 10(11)

The least common multiple (LCM) of 5, 3, 4, and 6 is 60. Multiplying both sides by 60 simplifies the equation into integer terms.

3

Distribute the coefficients to remove parentheses, taking care with the negative signs.

72x - 96 - 40x + 10 = 15x + 75 - 110

Distributing

-20

across

\left(2x - \frac{1}{2}\right)

yields

-40x + 10

, and distributing

24

across

(3x - 4)

yields

72x - 96

.

4

Combine like terms on both sides of the equation.

32x - 86 = 15x - 35

On the left side,

72x - 40x = 32x

and

-96 + 10 = -86

. On the right side,

75 - 110 = -35

.

5

Isolate the variable term

x

on one side and the constants on the other.

17x = 51 \implies x = 3

Subtracting

15x

from both sides gives

17x - 86 = -35

. Adding

86

to both sides gives

17x = 51

. Dividing by 17 yields

x = 3

.

6

Substitute

x = 3

into the requested expression

12x - 5

to find the final value.

12(3) - 5 = 31

The question asks for the value of the expression

12x - 5

rather than just the variable

x

.

Key Concept

Solving multi-step linear equations containing fractions by clearing denominators and distributing terms correctly.

Question 8Question →

A company allocates its monthly advertising budget between online advertisements and print media. Last month, the company spent $300$ more than half of its total budget on online advertisements, and $\frac{1}{6}$ of its total budget on print media. The remaining $700$ of the budget was spent on administrative fees. If $B$ represents the total budget in dollars, what is the value of $\frac{B}{10} + 150$ ?

270

300

450

405

315

Show answer & explanation

Answer: 450

Answer

The correct value is 450.

Evaluating the total budget equation yields

B = 3000

. Substituting this value into the expression

\frac{B}{10} + 150

gives

450

. This is correct because the individual allocations (online ads, print media, and administrative fees) sum to the total budget, and the fraction arithmetic is performed correctly.

Step-by-Step Solution

1

Define the variables and write the expressions for each category of expenses.

Let

B

be the total budget. Online advertisements budget is

\frac{1}{2}B + 300

, print media budget is

\frac{1}{6}B

, and administrative fees are

700

.

Translating verbal descriptions into algebraic expressions is necessary to build the equation.

2

Set up the linear equation representing the sum of all expenses equaling the total budget

B

.

B = \left(\frac{1}{2}B + 300\right) + \frac{1}{6}B + 700

The total budget is the sum of its individual components.

3

Group like terms and solve for the total budget

B

.

Combine constant terms:

300 + 700 = 1000

. Combine fraction terms:

\frac{1}{2}B + \frac{1}{6}B = \frac{3}{6}B + \frac{1}{6}B = \frac{4}{6}B = \frac{2}{3}B

. The equation becomes

B = \frac{2}{3}B + 1000

. Subtracting

\frac{2}{3}B

from both sides gives

\frac{1}{3}B = 1000

, which yields

B = 3000

.

This isolates the variable

B

to find the total budget value.

4

Evaluate the required expression

\frac{B}{10} + 150

using the solved value of

B

.

\frac{3000}{10} + 150 = 300 + 150 = 450

The question asks for the value of this specific expression rather than

B

itself.

Key Concept

Solving linear equations in one variable, including translating word problems with fractional terms and evaluating algebraic expressions.

Alternative Method

Instead of solving the equation algebraically, we could test the options to find the total budget

B

. Since each option represents the value of

V = \frac{B}{10} + 150

, we can express

B

as

B = 10(V - 150)

. For the correct option of

450

, we get

B = 10(450 - 150) = 3000

. Substituting

B = 3000

back into the original word problem description: half the budget plus

300

is

1500 + 300 = 1800

; one-sixth of the budget is

500

; the remaining is

3000 - 1800 - 500 = 700

, which matches the given administrative fees.

Estimated Time:2m 0s

Question 9Question →

An online retailer determines that the cost to ship a package of weight $w$ pounds is given by the linear expression $C(w) = kw + b$ , where $k$ and $b$ are constants. Shipping a $3$ -pound package costs $11.50$ dollars, and shipping an $8$ -pound package costs $24.00$ dollars. If the total shipping cost for two packages is $47.00$ dollars, and one of the packages weighs $5$ pounds, what is the weight, in pounds, of the other package?

Show answer & explanation

Answer: 10.6

Answer

The weight of the other package is 10.6 pounds.

The correct weight of 10.6 pounds is found by setting up a linear cost function

C(w) = 2.5w + 4

using the two data points, calculating the cost of the 5-pound package as

16.50

dollars, subtracting this from the total cost of

47.00

dollars to get

30.50

dollars, and solving

2.5w + 4 = 30.50

for the weight.

Step-by-Step Solution

1

Set up the linear system from the given costs

3k + b = 11.50

and

8k + b = 24.00

To determine the relationship between weight and shipping cost.

2

Solve for the slope

k

k = 2.50

Subtracting the first equation from the second eliminates

b

.

3

Solve for the intercept

b

b = 4.00

Substitute

k = 2.50

back into the first equation.

4

Determine the cost of the 5-pound package

C(5) = 16.50

dollars

Evaluate the linear expression

2.50(5) + 4.00

.

5

Determine the remaining cost for the second package

C(w_2) = 30.50

dollars

Subtract the cost of the first package from the total cost (

47.00 - 16.50

).

6

Solve the linear equation for the second package's weight

w_2 = 10.6

Solve

2.50w_2 + 4.00 = 30.50

for

w_2

.

Key Concept

Solving Linear Equations

Question 10Question →

If $x$ satisfies the equation $\frac{3(x - 2)}{5} - \frac{2x - 1}{3} = \frac{1}{15}$ , what is the value of the expression $2x + 7$ ?

Show answer & explanation

Answer: -21

Answer

The final value of the expression is

-21

.

Solving the equation by first multiplying both sides by the least common denominator of 15 yields the simplified equation

9(x - 2) - 5(2x - 1) = 1

. Distributing and combining terms yields

-x - 13 = 1

, which gives

x = -14

. Substituting this value into the expression

2x + 7

yields

-21

.

Step-by-Step Solution

1

Multiply the entire equation by the least common multiple of the denominators (

15

) to eliminate all fractions.

9(x - 2) - 5(2x - 1) = 1

Multiplying by the least common multiple of 5 and 3 eliminates the fractions and simplifies the equation.

2

Distribute the coefficients (

9

and

-5

) to their respective terms inside the parentheses.

9x - 18 - 10x + 5 = 1

Distributive property allows us to remove parentheses. Note that distributing

-5

to

-1

results in

+5

.

3

Combine the variable terms (

9x

and

-10x

) and the constant terms (

-18

and

5

) on the left side of the equation.

-x - 13 = 1

Combining like terms simplifies the expression to prepare for isolating the variable.

4

Isolate the variable term

-x

by adding

13

to both sides, then solve for

x

by multiplying by

-1

.

x = -14

Adding

13

yields

-x = 14

, and multiplying by

-1

isolates

x

to find its value.

5

Substitute

x = -14

into the given expression

2x + 7

.

2(-14) + 7 = -21

The question asks for the value of the expression

2x + 7

, not just

x

itself.

Key Concept

Solving multi-step linear equations involving fractions, distributive property with negative signs, and evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the least common multiple first, you can separate the fractions:

\frac{3}{5}x - \frac{6}{5} - \frac{2}{3}x + \frac{1}{3} = \frac{1}{15}

. Combining the

x

terms gives

(\frac{9}{15} - \frac{10}{15})x = -\frac{1}{15}x

. Combining the constant terms gives

-\frac{18}{15} + \frac{5}{15} = -\frac{13}{15}

. The equation becomes

-\frac{1}{15}x - \frac{13}{15} = \frac{1}{15}

. Multiplying the entire equation by

15

yields

-x - 13 = 1

, which gives

x = -14

, and substituting into

2x + 7

yields

-21

.

Estimated Time:2m 0s

Question 11Question →

For what value of $y$ is the equation $3(y - 4) = 5y + 2$ true?

Show answer & explanation

Answer: -7

Answer

The value of

y

that satisfies the equation is

-7

.

Distributing the 3 yields

3y - 12 = 5y + 2

. Subtracting

3y

from both sides gives

-12 = 2y + 2

. Subtracting 2 from both sides gives

-14 = 2y

. Dividing by 2 results in

y = -7

.

Step-by-Step Solution

1

Distribute the 3 on the left side of the equation.

3y - 12 = 5y + 2

To simplify the expression by expanding the parentheses.

2

Subtract

3y

from both sides of the equation.

-12 = 2y + 2

To collect the variable terms on the right side of the equation.

3

Subtract 2 from both sides of the equation.

-14 = 2y

To isolate the variable term.

4

Divide both sides by 2.

y = -7

To solve for

y

.

Key Concept

Solving linear equations by distributing and isolating the variable.

Estimated Time:45s

Question 12Question →

If $\frac{3}{5}x - 7 = 8$ , what is the value of $2x - 3$ ?

15

25

44

47

\frac{1}{3}

Show answer & explanation

Answer: 47

Answer

47

To find the value of

2x - 3

, first solve the equation

\frac{3}{5}x - 7 = 8

for

x

. Adding 7 to both sides of the equation gives

\frac{3}{5}x = 15

. Multiplying both sides by the reciprocal

\frac{5}{3}

yields

x = 25

. Finally, substitute

25

for

x

in the expression

2x - 3

to get

2(25) - 3 = 50 - 3 = 47

.

Step-by-Step Solution

1

Add 7 to both sides of the equation

\frac{3}{5}x - 7 = 8

to isolate the variable term.

\frac{3}{5}x = 15

Adding 7 to both sides eliminates the constant on the left side of the equation.

2

Multiply both sides of the equation by

\frac{5}{3}

to solve for

x

.

x = 25

Multiplying by the reciprocal of

\frac{3}{5}

isolates

x

on the left side of the equation.

3

Substitute

x = 25

into the expression

2x - 3

.

2(25) - 3 = 50 - 3 = 47

The question asks for the value of the expression

2x - 3

rather than just the variable

x

.

Key Concept

Solving two-step linear equations involving fractions, and evaluating algebraic expressions.

Alternative Method

Instead of solving for

x

first, you can express the target expression in terms of

\frac{3}{5}x

. Specifically, note that

2x - 3 = \frac{10}{3}(\frac{3}{5}x) - 3

. Since

\frac{3}{5}x = 15

, substituting this directly gives

\frac{10}{3}(15) - 3 = 50 - 3 = 47

.

Estimated Time:45s

Question 13Question →

A newly designed temperature scale, Scale X, is related to the Celsius scale ( $^{\circ}\text{C}$ ) by a linear equation. Water freezes at $0^{\circ}\text{C}$ , which corresponds to $-15^{\circ}\text{X}$ , and water boils at $100^{\circ}\text{C}$ , which corresponds to $135^{\circ}\text{X}$ . If a chemical reaction must be maintained at a temperature where the reading on Scale X is exactly $2.5$ times the reading on the Celsius scale, what is this temperature in degrees Celsius?

-11.5

-8.2

-15.0

13.6

15.0

Show answer & explanation

Answer: $-15.0$

Answer

-15.0

degrees Celsius

By writing the linear relationship between Scale X (

X

) and Celsius (

C

) as

X = mC + k

, we determine the constants using the given coordinates:

(0, -15)

yields

k = -15

, and

(100, 135)

yields

m = 1.5

. The resulting equation is

X = 1.5C - 15

. We then substitute the given condition

X = 2.5C

, resulting in

2.5C = 1.5C - 15

. Subtracting

1.5C

from both sides gives

C = -15

.

Step-by-Step Solution

1

Set up the general linear equation relating Scale X (

X

) and Celsius (

C

).

X = mC + k

Since the relationship is linear, it can be represented by a slope-intercept linear model.

2

Use the freezing point of water to find the y-intercept

k

.

When

C = 0

,

X = -15

, so

-15 = m(0) + k \implies k = -15

.

The freezing point of water provides the point

(0, -15)

on the linear graph.

3

Use the boiling point of water to find the slope

m

.

When

C = 100

,

X = 135

, so

135 = m(100) - 15 \implies 150 = 100m \implies m = 1.5

.

The boiling point of water provides the second point

(100, 135)

to determine the rate of change.

4

Substitute the condition

X = 2.5C

into the linear equation and solve for

C

.

2.5C = 1.5C - 15 \implies 1.0C = -15 \implies C = -15

.

This isolates the Celsius variable to find the temperature where the Scale X value is exactly

2.5

times the Celsius value.

Key Concept

Formulating and solving a linear equation from word-problem constraints and coordinate pairs.

Alternative Method

Instead of deriving the full equation, you can test the options directly. For example, check

-15^{\circ}\text{C}

. The distance from freezing (

0^{\circ}\text{C}

) to

-15^{\circ}\text{C}

is

-15

units. Since Scale X changes by

1.5

units for every

1

unit of Celsius (calculated from a change of

150

on Scale X for

100

on Celsius), Scale X will change by

1.5 \times (-15) = -22.5

units from its freezing point value of

-15

. This yields

-15 - 22.5 = -37.5^{\circ}\text{X}

. Checking the ratio:

\frac{-37.5}{-15} = 2.5

, which matches the given condition.

Estimated Time:2m 0s

Question 14Question →

For what value of $k$ does the equation $\frac{1}{5}(kx - 3) - \frac{1}{3}(2x - 5) = 2$ have a solution of $x = 7$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Substituting

x = 7

reduces the equation to

\frac{1}{5}(7k - 3) - 3 = 2

. Adding 3 to both sides yields

\frac{1}{5}(7k - 3) = 5

. Multiplying by 5 gives

7k - 3 = 25

. Adding 3 gives

7k = 28

, which results in

k = 4

.

Step-by-Step Solution

1

Substitute

x = 7

into the equation.

\frac{1}{5}(7k - 3) - \frac{1}{3}(2(7) - 5) = 2

Since

x = 7

is given as a solution, substituting it into the equation must make the equality true.

2

Evaluate and simplify the expression in the second term.

\frac{1}{3}(14 - 5) = \frac{1}{3}(9) = 3

Follow the order of operations by simplifying the expression inside the parentheses first.

3

Isolate the fractional term containing the variable

k

.

\frac{1}{5}(7k - 3) - 3 = 2 \implies \frac{1}{5}(7k - 3) = 5

Add 3 to both sides of the equation to eliminate the subtraction of 3.

4

Clear the fraction and solve the remaining linear equation for

k

.

7k - 3 = 25 \implies 7k = 28 \implies k = 4

Multiply both sides by 5 to eliminate the denominator, add 3 to isolate the term with

k

, and divide by 7.

Key Concept

Solving multi-step linear equations containing parameters and fractions.

Estimated Time:1m 30s

Question 15Question →

For a certain value of $x$ , the expression $\frac{2}{5}x + 3$ is equal to $11$ . What is the value of $2x + 5$ ?

11.4

21

27

45

85

Show answer & explanation

Answer: 45

Answer

45

To find the value of

2x + 5

, we first isolate the variable

x

in the equation

\frac{2}{5}x + 3 = 11

. Subtracting

3

from both sides gives

\frac{2}{5}x = 8

. Multiplying both sides by the reciprocal of the coefficient,

\frac{5}{2}

, yields

x = 8 \times \frac{5}{2} = 20

. Finally, substituting

x = 20

into the expression

2x + 5

gives

2(20) + 5 = 40 + 5 = 45

.

Step-by-Step Solution

1

Set up the equation based on the text and subtract 3 from both sides of the equation to isolate the variable term.

\frac{2}{5}x = 8

To solve for

x

, we first need to isolate the term containing

x

.

2

Multiply both sides of the equation by

\frac{5}{2}

to solve for

x

.

x = 20

Multiplying by the reciprocal of the coefficient cancels the fraction, leaving

x

isolated.

3

Substitute

x = 20

into the expression

2x + 5

.

2(20) + 5 = 45

The question asks for the value of

2x + 5

, so we evaluate it using the solved value of

x

.

Key Concept

Solving Linear Equations

Alternative Method

We can also multiply the entire equation by

5

first to eliminate the fraction:

2x + 15 = 55

. Subtracting

15

from both sides gives

2x = 40

. Since the target expression is

2x + 5

, we can simply add

5

to both sides of

2x = 40

to get

2x + 5 = 45

, avoiding the need to solve for

x

directly.

Estimated Time:45s

Question 16Question →

If $2.5(x - 4) = 15$ , what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The value of

x

is

10

.

To solve the equation

2.5(x - 4) = 15

, divide both sides by

2.5

to obtain

x - 4 = 6

. Then, add

4

to both sides to get the final solution of

x = 10

. Alternatively, distribute

2.5

to get

2.5x - 10 = 15

, add

10

to both sides to get

2.5x = 25

, and divide by

2.5

to get

x = 10

.

Step-by-Step Solution

1

Divide both sides of the equation by

2.5

.

x - 4 = 6

To isolate the parenthetical term on the left side of the equation.

2

Add

4

to both sides of the equation.

x = 10

To isolate the variable

x

.

Key Concept

Solving multi-step linear equations using inverse operations.

Alternative Method

Distribute

2.5

to get

2.5x - 10 = 15

. Add

10

to both sides of the equation to get

2.5x = 25

. Divide both sides by

2.5

to find

x = 10

.

Estimated Time:45s

Question 17Question →

A chemist is preparing a mixture. The volume of acid, $V$ in liters, required for a specific reaction satisfies the equation:

\frac{3}{5}(2V - 7) - \frac{1}{2}(V + 4) = \frac{1}{10}(3V - 2)

What is the value of the expression $4V + 3$ ?

15

23

27

47

63

Show answer & explanation

Answer: 63

Answer

The value of the expression is 63.

The correct answer is 63. Multiplying both sides of the equation by the least common denominator of 10 eliminates the fractions and yields

6(2V - 7) - 5(V + 4) = 3V - 2

. Distributing the factors results in

12V - 42 - 5V - 20 = 3V - 2

, which simplifies to

7V - 62 = 3V - 2

. Subtracting

3V

and adding 62 to both sides produces

4V = 60

, which gives

V = 15

. Substituting 15 into the expression

4V + 3

results in

4(15) + 3 = 63

.

Step-by-Step Solution

1

Multiply the entire equation by the least common denominator (LCD) to eliminate the fractions.

6(2V - 7) - 5(V + 4) = 1(3V - 2)

The denominators are 5, 2, and 10, so the LCD is 10.

2

Distribute the constants on the left side of the equation.

12V - 42 - 5V - 20 = 3V - 2

Distributing 6 to

(2V - 7)

yields

12V - 42

, and distributing

-5

to

(V + 4)

yields

-5V - 20

.

3

Combine like terms on the left side of the equation.

7V - 62 = 3V - 2

Combining the variable terms gives

12V - 5V = 7V

, and combining the constant terms gives

-42 - 20 = -62

.

4

Isolate the variable term by performing inverse operations.

4V = 60

Subtract

3V

from both sides to get

4V

, and add 62 to both sides to get 60.

5

Solve for

V

by dividing both sides of the equation.

V = 15

Dividing 60 by 4 yields 15.

6

Substitute the value of

V

into the requested expression.

4(15) + 3 = 63

The question asks for the value of the expression

4V + 3

, not the value of

V

.

Key Concept

Solving multi-step linear equations containing fractions by finding a common denominator, distributing terms correctly, and evaluating algebraic expressions.

Estimated Time:2m 0s

Question 18Question →

A company offers two monthly data storage plans. Plan A costs $32.00 plus$ 0.08 per gigabyte of data stored. Plan B costs $15.00 plus$ 0.14 per gigabyte for the first 150 gigabytes of data stored, and $0.12 per gigabyte for all data stored beyond the first 150 gigabytes. For how many gigabytes of data stored in a single month would the monthly cost under both plans be exactly the same?

Show answer & explanation

Answer: 350

Answer

The monthly costs under both plans are exactly the same when the data stored is 350 gigabytes.

Equating the two cost functions yields

32 + 0.08g = 15 + 0.14(150) + 0.12(g - 150)

, which simplifies to

32 + 0.08g = 18 + 0.12g

. Solving for

g

gives

0.04g = 14

, which results in

g = 350

.

Step-by-Step Solution

1

Determine the cost function for Plan A.

C_A(g) = 32 + 0.08g

Plan A charges a flat fee of

32.00 and

0.08 per gigabyte.

2

Determine the cost function for Plan B when the usage exceeds 150 gigabytes.

C_B(g) = 15 + 0.14(150) + 0.12(g - 150) = 18 + 0.12g

Plan B charges

15.00 flat,

0.14 per gigabyte for the first 150 gigabytes, and $0.12 per gigabyte for any additional usage.

3

Equate the two cost functions and solve for

g

.

32 + 0.08g = 18 + 0.12g \implies 14 = 0.04g \implies g = 350

To find the usage where both plans cost the same, set their cost functions equal and isolate the variable

g

.

Key Concept

Solving linear equations in real-life contexts involving piecewise rates

Question 19Question →

The weight of Box A is $1.5$ pounds more than $\frac{2}{3}$ of the weight of Box B. If the weight of Box A is $13.5$ pounds, what is the weight of Box B, in pounds?

8

10.5

13.5

18

22.5

Show answer & explanation

Answer: 18

Answer

The weight of Box B is 18 pounds.

The weight of 18 pounds is correct. Translating the word problem yields the linear equation

A = \frac{2}{3}B + 1.5

. Substituting the weight of Box A (

13.5

) gives

13.5 = \frac{2}{3}B + 1.5

. Subtracting

1.5

from both sides results in

12 = \frac{2}{3}B

. Multiplying both sides by the reciprocal

\frac{3}{2}

isolates

B

, giving

B = 12 \times \frac{3}{2} = 18

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

Let

A

represent the weight of Box A and

B

represent the weight of Box B. The description translates to the equation:

A = \frac{2}{3}B + 1.5

.

This establishes the mathematical relationship between the weights of the two boxes.

2

Substitute the given value for Box A into the equation and isolate the term containing the variable B.

13.5 = \frac{2}{3}B + 1.5 \implies 12 = \frac{2}{3}B

.

Substituting

13.5

for

A

and subtracting

1.5

from both sides simplifies the equation to isolate the fraction term.

3

Solve for B by multiplying both sides of the equation by the reciprocal of the coefficient.

B = 12 \times \frac{3}{2} = 18

.

Multiplying by the reciprocal

\frac{3}{2}

isolates

B

to find its value.

Key Concept

Translating real-world descriptions into multi-step linear equations and solving them using inverse operations.

Estimated Time:1m 0s

Question 20Question →

A school club sold 20 rolls of wrapping paper and 25 boxes of greeting cards for a fundraiser. The price of a roll of wrapping paper was $\frac{3}{4}$ of the price of a box of greeting cards. If the club raised a total of $800 from these sales, what was the price of a single roll of wrapping paper?

$5.00

$20.00

$15.00

$20.65

$26.67

Show answer & explanation

Answer: $15.00

Answer

The price of a single roll of wrapping paper was $15.00.

The correct answer is

15.00. By representing the price of a roll of wrapping paper as

W

and the price of a box of greeting cards as

G

, the problem states that

W = \frac{3}{4}G

, which means

G = \frac{4}{3}W

. The total revenue equation is

20W + 25G = 800

. Substituting

G

gives

20W + 25\left(\frac{4}{3}W\right) = 800

, which simplifies to

20W + \frac{100}{3}W = 800

. Multiplying the entire equation by 3 to clear the fraction results in

60W + 100W = 2400

. Combining like terms gives

160W = 2400

, which yields

W = 15$.

Step-by-Step Solution

1

Define variables for the unknowns and translate the price relationship into an algebraic equation.

Let

W

be the price of a roll of wrapping paper and

G

be the price of a box of greeting cards. The relationship is given by

W = \frac{3}{4}G

, which can be rearranged to express

G

in terms of

W

:

G = \frac{4}{3}W

.

Expressing one variable in terms of another allows us to set up a single-variable linear equation.

2

Write the linear equation representing the total revenue from the fundraiser sales.

The total revenue from selling 20 rolls of wrapping paper and 25 boxes of greeting cards is

20W + 25G = 800

. Substituting

G = \frac{4}{3}W

gives:

20W + 25\left(\frac{4}{3}W\right) = 800

, which simplifies to

20W + \frac{100}{3}W = 800

.

This sets up the equation that we need to solve to find the value of

W

.

3

Clear the fraction by multiplying all terms by 3 and solve for

W

.

Multiplying the entire equation by 3 yields:

3(20W) + 3\left(\frac{100}{3}W\right) = 3(800) \implies 60W + 100W = 2400

. Combining like terms gives

160W = 2400

. Dividing by 160 yields

W = 15

.

Clearing the denominator simplifies the equation to a standard linear form that can be solved directly.

Key Concept

Solving linear equations derived from real-world contexts, particularly those involving fractional relationships and multi-step isolation.

Alternative Method

Instead of expressing

G

in terms of

W

first, solve for

G

directly by substituting

W = \frac{3}{4}G

into the revenue equation. This gives

20\left(\frac{3}{4}G\right) + 25G = 800 \implies 15G + 25G = 800 \implies 40G = 800 \implies G = 20

. Then, calculate

W = \frac{3}{4}(20) = 15

.

Estimated Time:2m 30s

Solving Linear Equations

Answer

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Key Concept

Common Mistakes

Alternative Method

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Answer

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