Solving Linear Equations Practice Questions — ACT — Page 2

Question 21Question →

A storage tank contains $18$ gallons of water and is being filled at a constant rate of $\frac{2}{3}$ gallons per minute. A second storage tank contains $33$ gallons of water and is being drained at a constant rate of $\frac{5}{6}$ gallons per minute. If both tanks begin their processes at the same time, they will contain the same amount of water after $m$ minutes. What is the value of $2m + 3$ ?

23

26

33

71

183

Show answer & explanation

Answer: 23

Answer

23

The correct answer is 23. By representing the volume of the first tank as

18 + \frac{2}{3}m

and the second tank as

33 - \frac{5}{6}m

, setting them equal gives

18 + \frac{2}{3}m = 33 - \frac{5}{6}m

. Solving this equation by finding a common denominator for the fraction coefficients yields

\frac{9}{6}m = 15

, which simplifies to

\frac{3}{2}m = 15

, and thus

m = 10

. Substituting

m = 10

into the expression

2m + 3

results in

2(10) + 3 = 23

.

Step-by-Step Solution

1

Set up the equation representing the water volume in both tanks over time.

18 + \frac{2}{3}m = 33 - \frac{5}{6}m

The first tank starts with

18

gallons and increases by

\frac{2}{3}

gallons per minute, while the second starts with

33

gallons and decreases by

\frac{5}{6}

gallons per minute. We set their volumes equal to find when they contain the same amount.

2

Isolate the variable terms on one side and the constant terms on the other side.

\frac{2}{3}m + \frac{5}{6}m = 33 - 18

Adding

\frac{5}{6}m

to both sides and subtracting

18

from both sides groups like terms together.

3

Find a common denominator to add the fraction coefficients.

\frac{4}{6}m + \frac{5}{6}m = 15 \Rightarrow \frac{9}{6}m = 15 \Rightarrow \frac{3}{2}m = 15

A common denominator of

6

is used to add the fractions

\frac{2}{3}

and

\frac{5}{6}

.

4

Solve for

m

by multiplying both sides by the reciprocal of the coefficient.

m = 15 \times \frac{2}{3} = 10

Multiplying by

\frac{2}{3}

isolates

m

on the left side.

5

Evaluate the expression

2m + 3

using the value of

m

.

2(10) + 3 = 20 + 3 = 23

Substitute

m = 10

into the expression and follow the correct order of operations (multiply first, then add).

Key Concept

Solving linear equations with fractional coefficients and translating word problems into algebraic equations.

Question 22Question →

If $\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}$ , what is the value of $3x + 2$ ?

Show answer & explanation

Answer: 20

Answer

The correct answer is 20, which is the value of the expression

3x + 2

when

x = 6

.

The value of the expression

3x + 2

is 20. Solving the linear equation

\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}

yields

x = 6

. Substituting

x = 6

into

3x + 2

gives

3(6) + 2 = 20

.

Step-by-Step Solution

1

Multiply the entire equation by the least common multiple of the denominators (12).

8(3x - 4) - 3(x + 2) = 88

This clears the fractions to make solving the equation simpler.

2

Distribute the coefficients to remove parentheses.

24x - 32 - 3x - 6 = 88

This allows like terms to be grouped together.

3

Combine like terms on the left side of the equation.

21x - 38 = 88

Simplifies the equation to prepare for isolating the variable.

4

Add 38 to both sides of the equation.

21x = 126

Isolates the variable term on one side of the equation.

5

Divide both sides by 21 to solve for

x

.

x = 6

Finds the value of the variable

x

.

6

Substitute

x = 6

into the target expression

3x + 2

.

3(6) + 2 = 20

Calculates the final requested value.

Key Concept

Solving multi-step linear equations with fractions and evaluating algebraic expressions.

Question 23Question →

For a certain real number $x$ , the equation $\frac{3}{4}(x - 3) - \frac{1}{3}(2x + 5) = -2$ is true. What is the value of $2x - 5$ ?

Show answer & explanation

Answer: 41

Answer

The value of the expression

2x - 5

is 41.

Solving the linear equation by multiplying both sides by the least common denominator of 12 yields the simplified equation

9(x - 3) - 4(2x + 5) = -24

. Expanding the terms gives

9x - 27 - 8x - 20 = -24

, which simplifies to

x - 47 = -24

. Adding 47 to both sides gives

x = 23

. Finally, evaluating the expression

2x - 5

for

x = 23

results in

2(23) - 5 = 41

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator of 12 to eliminate fractions.

9(x - 3) - 4(2x + 5) = -24

Multiplying by the LCD clears all fractional coefficients, making the equation easier to solve.

2

Distribute the constants and expand the terms on the left side of the equation.

9x - 27 - 8x - 20 = -24

Applying the distributive property removes the parentheses.

3

Combine like terms on the left side of the equation.

x - 47 = -24

Simplifying the equation makes it easier to isolate the variable

x

.

4

Isolate the variable

x

by adding 47 to both sides of the equation.

x = 23

This determines the value of the unknown variable

x

.

5

Substitute

x = 23

into the target expression

2x - 5

.

41

The question asks for the value of the expression

2x - 5

, not just the value of

x

.

Key Concept

Solving multi-step linear equations with fractional coefficients by clearing the denominators and then evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the LCD first, distribute the fractions directly:

\frac{3}{4}x - \frac{9}{4} - \frac{2}{3}x - \frac{5}{3} = -2

. Combine the

x

terms:

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

. Combine the constant terms:

-\frac{9}{4} - \frac{5}{3} = -\frac{27}{12} - \frac{20}{12} = -\frac{47}{12}

. This gives the equation

\frac{1}{12}x - \frac{47}{12} = -2

. Add

\frac{47}{12}

to both sides:

\frac{1}{12}x = -2 + \frac{47}{12} = -\frac{24}{12} + \frac{47}{12} = \frac{23}{12}

. Multiply by 12 to get

x = 23

, then evaluate

2x - 5 = 41

.

Estimated Time:1m 30s

Question 24Question →

In a chemistry laboratory, a beaker contains a mixture of water and acid. The volume of water in the beaker is $3$ liters more than twice the volume of acid. After $5$ liters of water are added to the beaker, the ratio of the volume of water to the volume of acid is $5$ to $2$ . If no acid was added or removed, what was the initial volume of water, in liters, in the beaker?

5

16

19

35

50

Show answer & explanation

Answer: 35

Answer

35

The correct answer of 35 liters is found by setting up the linear equation where the initial volume of acid is

a

and the initial volume of water is

2a + 3

. Adding 5 liters of water gives

2a + 8

liters of water. Setting the ratio of water to acid to

\frac{5}{2}

and solving gives

a = 16

. Finally, substituting

a = 16

into the expression for the initial volume of water (

2a + 3

) yields 35 liters.

Step-by-Step Solution

1

Define variables for the initial volumes of acid and water based on the given relationships.

Let

a

be the initial volume of acid in liters. The initial volume of water is

2a + 3

liters.

This translates the statement 'the volume of water is 3 liters more than twice the volume of acid' into algebraic expressions.

2

Set up a linear equation representing the ratio after adding 5 liters of water.

The new volume of water is

(2a + 3) + 5 = 2a + 8

liters. The ratio of water to acid is

\frac{2a + 8}{a} = \frac{5}{2}

.

This uses the new state of the mixture to form an equation that can be solved for

a

.

3

Solve the linear equation for

a

by cross-multiplying and isolating the variable.

2(2a + 8) = 5a \implies 4a + 16 = 5a \implies a = 16

.

Cross-multiplication removes the fractions and allows for standard term isolation.

4

Calculate the initial volume of water.

Initial water volume =

2a + 3 = 2(16) + 3 = 35

liters.

The question asks for the initial volume of water, which is represented by

2a + 3

, not the volume of acid

a

.

Key Concept

Solving linear equations in one variable derived from word problems.

Alternative Method

Instead of solving algebraically, one could test the answer choices. For example, testing 35 liters of water means the initial acid is

(35 - 3) / 2 = 16

liters. Adding 5 liters of water gives 40 liters of water. The ratio of water to acid is

40 : 16

, which simplifies to

5 : 2

. This confirms 35 is correct.

Estimated Time:1m 30s

Question 25Question →

If the equation $\frac{2}{5}(x - 4) + \frac{1}{3}(2x + k) = 7$ is true when $x = 9$ , what is the value of $k$ ?

-17

1

-3

3

8

Show answer & explanation

Answer: $-3$

Answer

-3

Substituting

x = 9

into the equation gives

\frac{2}{5}(9 - 4) + \frac{1}{3}(2(9) + k) = 7

. Simplifying the terms yields

2 + \frac{18 + k}{3} = 7

. Subtracting 2 from both sides results in

\frac{18 + k}{3} = 5

. Multiplying by 3 gives

18 + k = 15

. Subtracting 18 from both sides gives

k = -3

.

Step-by-Step Solution

1

Substitute

x = 9

into the given equation.

\frac{2}{5}(9 - 4) + \frac{1}{3}(2(9) + k) = 7

We are given that the equation is true when

x = 9

.

2

Simplify the operations inside the parentheses.

2 + \frac{18 + k}{3} = 7

Simplifying

\frac{2}{5}(5)

gives 2, and

2(9)

gives 18.

3

Subtract 2 from both sides of the equation.

\frac{18 + k}{3} = 5

Isolating the fraction term simplifies the equation.

4

Multiply both sides by 3.

18 + k = 15

Clearing the denominator allows us to isolate the variable

k

.

5

Subtract 18 from both sides of the equation.

k = -3

This isolates

k

to find its value.

Key Concept

Solving linear equations in one variable by substitution and simplification

Question 26Question →

A landscaping company purchases bags of grass seed for $30$ each and bags of fertilizer for $18$ each. For a large project, the company purchased $2$ fewer than $1.5$ times as many bags of fertilizer as bags of grass seed. If the total cost of the grass seed and fertilizer was $420$ , how many bags of fertilizer did the company purchase?

Show answer & explanation

Answer: 10

Answer

The company purchased 10 bags of fertilizer.

The correct answer is 10. By writing the number of fertilizer bags as

f = 1.5g - 2

and substituting this expression into the total cost equation

30g + 18f = 420

, we get

30g + 18(1.5g - 2) = 420

. Simplifying the expression leads to

57g - 36 = 420

, which yields

g = 8

. Substituting

g = 8

back into the relationship for

f

gives

f = 1.5(8) - 2 = 10

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

g

represent the number of grass seed bags and

f

represent the number of fertilizer bags.

Setting up variables is the first step in translating the word problem into solvable algebraic equations.

2

Translate the relationship between the quantities of bags into an equation.

f = 1.5g - 2

The problem states that the number of fertilizer bags purchased is 2 fewer than 1.5 times the number of grass seed bags purchased.

3

Write the total cost equation using the prices and variables.

30g + 18f = 420

The total cost of

420

is the sum of the cost of the grass seed (

30

per bag) and the fertilizer (

18

per bag).

4

Substitute the equation from Step 2 into the cost equation from Step 3.

30g + 18(1.5g - 2) = 420

Substituting allows us to solve a single linear equation with one variable.

5

Distribute the 18 through the parentheses and combine like terms.

57g - 36 = 420

Distributing gives

18 \times 1.5g = 27g

and

18 \times -2 = -36

. Combining

30g

and

27g

yields

57g

.

6

Isolate the variable term by adding 36 to both sides of the equation.

57g = 456

To solve for

g

, we must isolate the term containing the variable.

7

Divide both sides by 57 to find the value of

g

.

g = 8

This division yields the number of grass seed bags purchased.

8

Substitute the value of

g

back into the relation for

f

to find the final answer.

f = 1.5(8) - 2 = 10

The question asks for the number of fertilizer bags (

f

), not grass seed bags (

g

).

Key Concept

Solving a linear equation by substitution and distributing across linear terms.

Estimated Time:1m 30s

Question 27Question →

For a certain real number $x$ , the equation $0.4(3x - 5) - 0.15(2x + 8) = 1.3$ is true. What is the value of $4x - 3$ ?

Show answer & explanation

Answer: 17

Answer

The value of the expression is 17.

Distributing the decimals results in

1.2x - 2 - 0.3x - 1.2 = 1.3

. Combining like terms yields

0.9x - 3.2 = 1.3

. Adding

3.2

to both sides results in

0.9x = 4.5

, which simplifies to

x = 5

after dividing by

0.9

. Substituting

x = 5

into the expression

4x - 3

gives

4(5) - 3 = 17

.

Step-by-Step Solution

1

Distribute the decimal factors through the parentheses on the left side of the equation.

1.2x - 2 - 0.3x - 1.2 = 1.3

To eliminate the parentheses and set up terms for simplification.

2

Combine the variable terms and constant terms on the left side of the equation.

0.9x - 3.2 = 1.3

To group like terms and simplify the equation.

3

Isolate the variable term by adding

3.2

to both sides of the equation.

0.9x = 4.5

To gather all constant terms on the right side of the equation.

4

Divide both sides of the equation by

0.9

to solve for

x

.

x = 5

To find the numerical value of the variable.

5

Substitute the value of

x

into the requested expression

4x - 3

.

4(5) - 3 = 17

To evaluate the specific expression requested by the question.

Key Concept

Solving multi-step linear equations with decimals and evaluating algebraic expressions

Question 28Question →

A fitness tracker records a user's daily steps. On Monday, the user walked a certain number of steps. On Tuesday, they walked $1.5$ times the number of steps they walked on Monday. On Wednesday, they walked $2,500$ fewer steps than they did on Tuesday. If the user walked a total of $21,500$ steps over these three days, how many steps did they walk on Monday?

4,750

6,000

6,500

8,000

9,000

Show answer & explanation

Answer: 6,000

Answer

6,000

The correct answer is

6,000

. By letting

x

represent the steps on Monday, we express Tuesday's steps as

1.5x

and Wednesday's steps as

1.5x - 2,500

. Summing these expressions gives the equation

x + 1.5x + 1.5x - 2,500 = 21,500

. Combining like terms yields

4x - 2,500 = 21,500

. Adding

2,500

to both sides gives

4x = 24,000

, and dividing by

4

results in

x = 6,000

.

Step-by-Step Solution

1

Define the variable and write expressions for each day's steps.

Let

x

be the number of steps walked on Monday. Tuesday's steps are

1.5x

, and Wednesday's steps are

1.5x - 2,500

.

This translates the word problem statements into algebraic terms using a single variable.

2

Set up the linear equation representing the total steps.

x + 1.5x + (1.5x - 2,500) = 21,500

The sum of the steps over the three days must equal the given total of

21,500

steps.

3

Combine like terms and solve for

x

.

4x - 2,500 = 21,500 \Rightarrow 4x = 24,000 \Rightarrow x = 6,000

Simplifying the equation isolates the variable to find Monday's step count.

Key Concept

Formulating and solving a single-variable linear equation from a word problem context.

Estimated Time:1m 15s

Question 29Question →

A delivery drone starts a flight with a battery charge of $95\%$ . The battery charge decreases at a constant rate of $1.5\%$ per minute of flight time. During the flight, the drone lands on a charging station for $15$ minutes, during which its battery charge increases at a constant rate of $2.4\%$ per minute. After this charging period, the drone resumes its flight. If the drone ends its flight with a battery charge of $65\%$ , and its total flight time (excluding the time spent charging) was $t$ minutes, what is the value of $t$ ?

Show answer & explanation

Answer: 44

Answer

44

The correct answer is found by setting up a linear equation that models the drone's battery charge changes. The drone starts with

95\%

battery, loses

1.5\%

per minute for

t

minutes, and gains

36\%

from charging (

15 \times 2.4\%

). Setting this equal to the final charge of

65\%

gives the equation

95 - 1.5t + 36 = 65

. Simplifying gives

131 - 1.5t = 65

, which leads to

-1.5t = -66

and

t = 44

.

Step-by-Step Solution

1

Calculate the total percentage of battery charge gained while charging.

36\%

The drone charges at a rate of

2.4\%

per minute for

15

minutes, so the total gain is

15 \times 2.4\% = 36\%

.

2

Set up a linear equation for the final battery charge.

95 - 1.5t + 36 = 65

The final battery charge (

65\%

) is the initial charge (

95\%

) minus the battery consumed during flight (

1.5\%

per minute for

t

minutes) plus the charge gained (

36\%

).

3

Combine constant terms on the left side of the equation.

131 - 1.5t = 65

Adding

95

and

36

simplifies the expression on the left side.

4

Isolate the variable term by subtracting

131

from both sides.

-1.5t = -66

Subtracting

131

from both sides leaves only the variable term on the left.

5

Divide both sides by

-1.5

to solve for

t

.

t = 44

Dividing

-66

by

-1.5

isolates

t

to find the total flight time.

Key Concept

Solving linear equations in a real-world context by setting up an algebraic equation.

Estimated Time:1m 30s

Question 30Question →

An online bookstore offers two subscription options for downloading e-books. Store A charges a monthly membership fee of $6.00$ plus $1.25$ per e-book download. Store B has no membership fee and charges $2.75$ per e-book download. A customer downloads the same number of e-books from each store in a single month and spends a total of $46.00$ across both stores. How many e-books did the customer download from each store?

4

7

10

11.5

13

Show answer & explanation

Answer: 10

Answer

10 e-books from each store

The correct answer represents the number of downloads from each store that satisfies the total cost equation. Let

x

represent the number of books downloaded from each store. The cost at Store A is

6.00 + 1.25x

and the cost at Store B is

2.75x

. Since the total spent is

46.00

, we write the equation

(6.00 + 1.25x) + 2.75x = 46.00

. Combining like terms gives

6.00 + 4.00x = 46.00

. Subtracting

6.00

from both sides gives

4.00x = 40.00

, and dividing by

4.00

yields

x = 10

.

Step-by-Step Solution

1

Define the variable and write the expressions for the cost of each store.

Let

x

be the number of e-books downloaded from each store. Store A cost is

6.00 + 1.25x

, and Store B cost is

2.75x

.

Establishing algebraic expressions represents the verbal statements mathematically.

2

Set up the total cost equation by summing the costs of both stores and setting it equal to the total spent.

(6.00 + 1.25x) + 2.75x = 46.00

The customer spends a combined total of

46.00

across both bookstores.

3

Combine like terms and solve for the variable

x

.

6.00 + 4.00x = 46.00 \implies 4.00x = 40.00 \implies x = 10

Isolating the variable determines the number of e-books downloaded from each store.

Key Concept

Solving linear equations in one variable derived from real-world scenarios.

Estimated Time:1m 30s

Question 31Question →

A technician uses a linear model to estimate the time, $t$ hours, required to complete a project. The estimate satisfies the equation:

\frac{2}{3}(t - 12) + \frac{5}{6}(t + 6) = 18

What is the value of $t$ ?

Show answer & explanation

Answer: 14

Answer

The value of

t

is

14

.

To solve the linear equation, we first multiply all terms by the least common multiple of the denominators, which is

6

. This simplifies the equation to

4(t - 12) + 5(t + 6) = 108

. Distributing the terms gives

4t - 48 + 5t + 30 = 108

. Combining like terms yields

9t - 18 = 108

. Adding

18

to both sides gives

9t = 126

. Finally, dividing by

9

gives

t = 14

.

Step-by-Step Solution

1

Multiply both sides of the equation by

6

to eliminate the denominators.

4(t - 12) + 5(t + 6) = 108

Multiplying by the least common multiple of the denominators clears the fractions, making the linear equation easier to solve.

2

Apply the distributive property to expand the terms.

4t - 48 + 5t + 30 = 108

Distributing the constants allows us to group like terms.

3

Combine the variable terms and the constant terms on the left side.

9t - 18 = 108

Simplifying the expression is a necessary step before isolating the variable.

4

Add

18

to both sides of the equation.

9t = 126

Adding the constant to both sides isolates the variable term on the left side.

5

Divide both sides of the equation by

9

to solve for

t

.

t = 14

Dividing by the coefficient of the variable yields the final solution.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and isolating the variable.

Question 32Question →

A local community garden allocates a plot of land for different types of vegetables. Tomatoes occupy $\frac{2}{5}$ of the total area, peppers occupy $\frac{1}{3}$ of the total area, and the remaining 80 square feet are planted with herbs. What is the total area, in square feet, of the community garden plot?

100

120

128

200

300

Show answer & explanation

Answer: 300

Answer

300 square feet

Let

x

be the total area of the garden plot. Since the garden is divided into tomatoes (

\frac{2}{5}

of the total area), peppers (

\frac{1}{3}

of the total area), and herbs (

80

square feet), the sum of these parts equals the total area:

\frac{2}{5}x + \frac{1}{3}x + 80 = x

. Finding a common denominator of 15 for the fractions gives

\frac{6}{15}x + \frac{5}{15}x + 80 = x

, which simplifies to

\frac{11}{15}x + 80 = x

. Subtracting

\frac{11}{15}x

from both sides yields

80 = \frac{4}{15}x

. Multiplying both sides by the reciprocal

\frac{15}{4}

gives

x = 80 \times \frac{15}{4} = 300

square feet.

Step-by-Step Solution

1

Define the variable and translate the given fractions and numbers into an equation. Let

x

represent the total area of the garden plot.

\frac{2}{5}x + \frac{1}{3}x + 80 = x

The sum of the areas of the tomato plot, the pepper plot, and the herb plot must equal the total area of the garden.

2

Find a common denominator to combine the fractional coefficients of

x

. The least common multiple of 5 and 3 is 15. Convert

\frac{2}{5}

and

\frac{1}{3}

to fractions with a denominator of 15.

\frac{6}{15}x + \frac{5}{15}x + 80 = x \implies \frac{11}{15}x + 80 = x

Combining like terms simplifies the expression to help isolate the variable.

3

Subtract

\frac{11}{15}x

from both sides of the equation to group all

x

terms on one side.

80 = x - \frac{11}{15}x \implies 80 = \frac{4}{15}x

This isolates the constant term on one side of the equation.

4

Multiply both sides of the equation by the reciprocal of

\frac{4}{15}

, which is

\frac{15}{4}

, to solve for

x

.

x = 80 \times \frac{15}{4} = 20 \times 15 = 300

This yields the value of the total area of the garden plot.

Key Concept

Solving linear equations in one variable with fractional coefficients

Question 33Question →

The length $L$ , in centimeters, of a copper rod at a temperature of $T$ degrees Celsius can be modeled by the linear equation $L = 150.04 + 0.012T$ . If the length of the rod is measured to be $150.40$ centimeters, what is its temperature in degrees Celsius?

Show answer & explanation

Answer: 30

Answer

The temperature of the rod is 30 degrees Celsius.

The correct answer is 30. By substituting the given length of

150.40

centimeters into the equation for

L

, we get

150.40 = 150.04 + 0.012T

. Subtracting

150.04

from both sides gives

0.36 = 0.012T

. Finally, dividing both sides by

0.012

yields the temperature

T = 30

degrees Celsius.

Step-by-Step Solution

1

Substitute the measured length

L = 150.40

into the equation.

150.40 = 150.04 + 0.012T

To set up the equation with the given value for length.

2

Subtract 150.04 from both sides of the equation.

0.36 = 0.012T

To isolate the variable term containing

T

.

3

Divide both sides of the equation by 0.012.

T = 30

To find the temperature

T

.

Key Concept

Solving a multi-step linear equation involving decimals

Question 34Question →

A marketing firm charges a flat setup fee of $160$ plus $3.20$ for each promotional brochure printed. For a large order, the firm applies a $15\%$ discount to the total cost (the sum of the setup fee and the printing cost). If the final discounted bill is $680$ , how many brochures were printed?

170

20

200

250

300

Show answer & explanation

Answer: 200

Answer

200

The correct answer of 200 brochures is found by formulating the equation

0.85(160 + 3.2b) = 680

. Dividing both sides by

0.85

yields

160 + 3.2b = 800

. Subtracting

160

gives

3.2b = 640

, and dividing by

3.2

results in

b = 200

.

Step-by-Step Solution

1

Translate the word problem into a linear equation where

b

represents the number of brochures. The total cost before discount is

160 + 3.2b

. Applying a

15\%

discount means paying

85\%

of this total.

0.85(160 + 3.2b) = 680

This sets up the relationship between the total discounted bill and the number of brochures.

2

Divide both sides of the equation by

0.85

to isolate the expression within parentheses.

160 + 3.2b = 800

Dividing by

0.85

simplifies the equation by calculating the total cost before the discount was applied.

3

Subtract

160

from both sides to isolate the term containing the variable

b

.

3.2b = 640

Subtracting the flat setup fee isolates the printing cost of the brochures.

4

Divide both sides by

3.2

to solve for

b

.

b = 200

Dividing by the per-brochure printing cost yields the total number of brochures printed.

Key Concept

Solving multi-step linear equations involving percentage discounts and decimal coefficients.

Alternative Method

Instead of dividing by

0.85

first, you can distribute

0.85

to both terms inside the parentheses to get

136 + 2.72b = 680

. Subtract

136

from both sides to get

2.72b = 544

, then divide by

2.72

to find

b = 200

.

Estimated Time:1m 30s

Question 35Question →

If $x$ is a real number that satisfies the equation $\frac{3}{5}(2x - 7) + 0.4 = 0.2(x + 3)$ , what is the value of $5x - 2$ ?

Show answer & explanation

Answer: 20

Answer

The value of the expression

5x - 2

is

20

.

First, convert the fraction

\frac{3}{5}

to the decimal

0.6

. The equation becomes

0.6(2x - 7) + 0.4 = 0.2(x + 3)

. Distribute on both sides to get

1.2x - 4.2 + 0.4 = 0.2x + 0.6

. Combine constant terms on the left side to get

1.2x - 3.8 = 0.2x + 0.6

. Subtract

0.2x

and add

3.8

to both sides to isolate the variable, resulting in

x = 4.4

. Finally, substitute

x = 4.4

into the expression

5x - 2

to get

5(4.4) - 2 = 22 - 2 = 20

.

Step-by-Step Solution

1

Convert the fraction and distribute the coefficients

1.2x - 4.2 + 0.4 = 0.2x + 0.6

To clear parentheses and align terms using decimals.

2

Combine constants on the left side

1.2x - 3.8 = 0.2x + 0.6

To simplify the left-hand side of the linear equation.

3

Isolate the variable

x

x = 4.4

To determine the value of the unknown variable.

4

Evaluate the target expression

20

To compute the final value of the expression

5x - 2

.

Key Concept

Solving linear equations with fractional and decimal coefficients

Question 36Question →

A laboratory technician mixes two solutions. The volume of the first solution is represented by $\frac{1}{2}(x - 5)$ liters, and the volume of the second solution is represented by $\frac{1}{3}(2x + 1)$ liters, where $x$ is a positive real number. If the sum of the volumes of these two solutions is $6$ liters, what is the value of $3x - 4$ ?

6

9

17

30

53

Show answer & explanation

Answer: $17$

Answer

The value of the expression is

17

.

The sum of the volumes of the two solutions is

6

liters, which translates to the linear equation

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

. Multiplying the entire equation by the least common multiple of the denominators,

6

, yields

3(x - 5) + 2(2x + 1) = 36

. Distributing the constants leads to

3x - 15 + 4x + 2 = 36

. Combining like terms gives

7x - 13 = 36

. Adding

13

to both sides results in

7x = 49

, which gives

x = 7

. Substituting

x = 7

into the expression

3x - 4

yields

3(7) - 4 = 17

, which is the correct value.

Step-by-Step Solution

1

Set up the linear equation based on the word problem context.

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

The sum of the volumes of the two solutions is given as

6

liters.

2

Clear the fractions by multiplying both sides of the equation by the least common multiple of the denominators.

3(x - 5) + 2(2x + 1) = 36

Multiplying by

6

eliminates the fractions and simplifies the equation for solving.

3

Distribute the coefficients to eliminate the parentheses.

3x - 15 + 4x + 2 = 36

Applying the distributive property allows like terms to be grouped.

4

Combine like terms and isolate the variable

x

.

7x - 13 = 36 \implies 7x = 49 \implies x = 7

Adding

13

to both sides and dividing by

7

solves for the value of the variable

x

.

5

Evaluate the expression requested in the question using the solved value of

x

.

3(7) - 4 = 21 - 4 = 17

Substituting

x = 7

into

3x - 4

yields the final requested value.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and evaluating variable expressions.

Alternative Method

Instead of clearing the fractions immediately, we can distribute the fractions first:

\frac{1}{2}x - 2.5 + \frac{2}{3}x + \frac{1}{3} = 6

. Converting to fractions with a common denominator of 6, we get

\frac{3}{6}x - \frac{15}{6} + \frac{4}{6}x + \frac{2}{6} = 6

, which simplifies to

\frac{7}{6}x - \frac{13}{6} = 6

. Adding

\frac{13}{6}

to both sides yields

\frac{7}{6}x = \frac{49}{6}

, so

7x = 49

, and

x = 7

. Substituting

x = 7

into the expression

3x - 4

yields

17

.

Estimated Time:1m 30s

Question 37Question →

A business analyst uses the linear equation $\frac{2}{5}(3p - 10) - 1.2 = 0.4(p + 5)$ to estimate the equilibrium price $p$ , in dollars, of a new product. What is the equilibrium price, in dollars, of the product?

Show answer & explanation

Answer: 9

Answer

The equilibrium price of the product is

9

dollars.

Converting the fraction to a decimal gives

0.4(3p - 10) - 1.2 = 0.4(p + 5)

. Dividing both sides by

0.4

results in

3p - 10 - 3 = p + 5

, which simplifies to

3p - 13 = p + 5

. Subtracting

p

and adding

13

to both sides gives

2p = 18

, and dividing by

2

results in

9

.

Step-by-Step Solution

1

Convert the fraction to decimal form

0.4(3p - 10) - 1.2 = 0.4(p + 5)

Converting

\frac{2}{5}

to

0.4

makes all terms decimals, simplifying further operations.

2

Divide both sides of the equation by

0.4

(3p - 10) - 3 = p + 5

Since

0.4

is a common factor and

1.2 / 0.4 = 3

, dividing both sides by

0.4

simplifies the coefficients.

3

Simplify the left side of the equation

3p - 13 = p + 5

Combine the constant terms

-10

and

-3

to simplify the expression.

4

Isolate the variable terms on one side

2p = 18

Subtract

p

from both sides and add

13

to both sides.

5

Solve for

p

p = 9

Divide both sides by

2

to find the final value.

Key Concept

Solving Linear Equations

Question 38Question →

A local coffee shop sells two types of coffee blends: House Blend and organic Reserve Blend. A bag of Reserve Blend costs $4.50$ dollars less than three times the cost of a bag of House Blend. A customer purchases $3$ bags of House Blend and $2$ bags of Reserve Blend for a total of $54.00$ dollars, excluding tax. What is the cost, in dollars, of a bag of Reserve Blend?

7.00

10.50

15.00

16.50

19.50

Show answer & explanation

Answer: 16.50

Answer

The cost of a bag of Reserve Blend is 16.50 dollars.

The cost of a bag of Reserve Blend is 16.50 dollars. Let

x

represent the cost of a bag of House Blend. A bag of Reserve Blend costs

3x - 4.50

. Since the total cost for

3

bags of House Blend and

2

bags of Reserve Blend is

54.00

, we write the linear equation

3x + 2(3x - 4.50) = 54

. Expanding the terms gives

3x + 6x - 9 = 54

, which simplifies to

9x - 9 = 54

. Adding

9

to both sides yields

9x = 63

. Dividing by

9

gives

x = 7.00

. Substituting

7.00

back into the Reserve Blend expression yields

3(7.00) - 4.50 = 16.50

dollars.

Step-by-Step Solution

1

Define variables for the costs of each blend.

Let

x

represent the cost of a bag of House Blend in dollars. The cost of a bag of Reserve Blend is represented by the expression

3x - 4.50

.

Establishing algebraic expressions for the unknowns based on the problem description allows us to set up a linear equation.

2

Set up the linear equation based on the total cost of the purchase.

The total cost of

3

bags of House Blend and

2

bags of Reserve Blend is

54.00

dollars:

3(x) + 2(3x - 4.50) = 54

.

The sum of the individual total costs of the two blends must equal the overall purchase total.

3

Distribute and combine like terms to simplify the equation.

3x + 6x - 9 = 54

simplifies to

9x - 9 = 54

.

Applying the distributive property removes the parentheses, allowing like terms to be combined.

4

Isolate the variable term by adding

9

to both sides.

9x = 63

.

To solve for

x

, we must first isolate the term containing the variable by performing the inverse operation.

5

Solve for

x

by dividing both sides by

9

.

x = 7.00

.

Dividing isolates

x

, giving the cost of a bag of House Blend.

6

Calculate the cost of a bag of Reserve Blend using the expression from Step 1.

3(7.00) - 4.50 = 21.00 - 4.50 = 16.50

dollars.

The question asks for the cost of a bag of Reserve Blend, not the House Blend, so we evaluate the expression

3x - 4.50

at

x = 7.00

.

Key Concept

Setting up and solving a single-variable linear equation to solve a real-world word problem with multiple unknown quantities.

Estimated Time:1m 30s

Question 39Question →

A shipping company charges a rate based on the weight of a package. The total cost $C$ , in dollars, to ship a package of weight $w$ pounds is given by the formula $C = \frac{5}{8}(w - 2) + 6.50$ for packages weighing more than $2$ pounds. If the shipping cost for a certain package is $\$14.00$ , what is the weight of the package, in pounds?

Show answer & explanation

Answer: 14

Answer

The weight of the package is 14 pounds.

Substituting

C = 14.00

into the formula gives

14.00 = \frac{5}{8}(w - 2) + 6.50

. Subtracting

6.50

from both sides yields

7.50 = \frac{5}{8}(w - 2)

. Multiplying both sides by the reciprocal

\frac{8}{5}

yields

12 = w - 2

. Finally, adding

2

to both sides gives the weight

w = 14

pounds.

Step-by-Step Solution

1

Substitute the total shipping cost into the formula.

14.00 = \frac{5}{8}(w - 2) + 6.50

Since the shipping cost

C

is given as

14.00

, we substitute this value into the formula to solve for the unknown weight

w

.

2

Subtract

6.50

from both sides of the equation.

7.50 = \frac{5}{8}(w - 2)

Subtracting

6.50

isolates the term containing the variable

w

on the right side of the equation.

3

Multiply both sides of the equation by the reciprocal of the fraction.

12 = w - 2

Multiplying by

\frac{8}{5}

eliminates the fractional coefficient of

\frac{5}{8}

on the right side.

4

Add

2

to both sides of the equation to solve for

w

.

w = 14

Adding

2

isolates the variable

w

, giving the final weight of the package.

Key Concept

Solving linear equations with fractional and decimal terms

Estimated Time:1m 30s

Question 40Question →

A craft cider company produces two specialty blends. The production cost, in dollars per gallon, of the premium blend is represented by the expression $\frac{3}{4}(d - 8)$ , where $d$ is the wholesale cost, in dollars, of a bushel of apples. The production cost, in dollars per gallon, of the dry blend is represented by the expression $0.2(2d + 5)$ . If the production cost per gallon is the same for both blends, what is this production cost, in dollars per gallon?

1.2

1.8

9

20

45

Show answer & explanation

Answer: 9

Answer

The production cost is 9 dollars per gallon.

The correct answer is the value obtained by setting the two cost expressions equal, solving for

d = 20

, and then substituting

d = 20

back into either expression to find the cost of

9

dollars per gallon.

Step-by-Step Solution

1

Set the two production cost expressions equal to each other.

\frac{3}{4}(d - 8) = 0.2(2d + 5)

The problem states that the production cost per gallon is the same for both blends, so their algebraic representations must be equal.

2

Express the decimal as a fraction and solve for

d

.

15(d - 8) = 4(2d + 5) \implies 15d - 120 = 8d + 20 \implies 7d = 140 \implies d = 20

Converting

0.2

to

\frac{1}{5}

allows us to clear fractions by multiplying both sides of the equation by the least common multiple of the denominators, which is

20

. We then distribute, isolate the variable terms, and solve for

d

.

3

Substitute the value of

d

back into either of the original cost expressions to find the production cost.

\text{Cost} = \frac{3}{4}(20 - 8) = \frac{3}{4}(12) = 9

The question asks for the production cost per gallon, not the value of the variable

d

. Substituting

d = 20

into the premium blend cost expression gives the final cost.

Key Concept

Solving linear equations involving fractions and decimals, and evaluating expressions using the solved variable value.

Alternative Method

Convert all numbers to decimals. The premium blend cost is

0.75(d - 8) = 0.75d - 6

. The dry blend cost is

0.2(2d + 5) = 0.4d + 1

. Equating them gives

0.75d - 6 = 0.4d + 1 \implies 0.35d = 7 \implies d = 20

. Then find the cost:

0.75(20) - 6 = 15 - 6 = 9

dollars.

Estimated Time:1m 30s

Solving Linear Equations

Answer

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Key Concept

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