Algebra Practice Questions — SAT

Question 1Question →

A florist sells carnations for $1.50$ each and roses for $3.00$ each. The equation $1.50c + 3.00r = 45.00$ represents the possible number of carnations, $c$ , and roses, $r$ , that a customer can buy for exactly $45.00$ . If the customer buys 8 carnations, how many roses can they buy?

Show answer & explanation

Answer: 11

Answer

11

To find the number of roses the customer can buy, substitute the number of carnations bought, which is 8, for

c

in the given equation. This yields

1.50(8) + 3.00r = 45.00

. Simplifying the product gives

12.00 + 3.00r = 45.00

. Subtracting 12.00 from both sides of the equation gives

3.00r = 33.00

. Finally, dividing both sides by 3.00 yields

r = 11

. Thus, the customer can buy 11 roses.

Step-by-Step Solution

1

Substitute the given value for carnations,

c = 8

, into the linear equation.

1.50(8) + 3.00r = 45.00

We are given that the customer buys 8 carnations, so

c

is replaced by 8 to solve for the number of roses,

r

.

2

Multiply

1.50

by

8

to simplify the term.

12.00 + 3.00r = 45.00

Simplifying the constant term on the left side of the equation.

3

Subtract

12.00

from both sides of the equation to isolate the term with

r

.

3.00r = 33.00

To solve for

r

, we must get the term

3.00r

by itself on one side.

4

Divide both sides of the equation by

3.00

to solve for

r

.

r = 11

Dividing isolates the variable

r

.

Key Concept

Solving a linear equation in two variables by substitution when one variable's value is known.

Question 2Question →

In the equation below, $a$ is a constant.

\frac{1}{3}(ax - 2) - \frac{1}{2}(x - 4a) = 3x - 5

If the solution to the equation is $x = 2$ , what is the value of $a$ ?

-2

\frac{1}{2}

1

2

Show answer & explanation

Answer: $1$

Answer

1

Substituting

x = 2

into the equation gives

\frac{1}{3}(2a - 2) - \frac{1}{2}(2 - 4a) = 3(2) - 5

, which simplifies to

\frac{1}{3}(2a - 2) - (1 - 2a) = 1

. Multiplying the entire equation by 3 to clear the fraction yields

(2a - 2) - 3(1 - 2a) = 3

. Distributing the negative 3 yields

2a - 2 - 3 + 6a = 3

, which simplifies to

8a - 5 = 3

. Adding 5 to both sides results in

8a = 8

, which gives

a = 1

.

Step-by-Step Solution

1

Substitute

x = 2

into the given equation.

\frac{1}{3}(2a - 2) - \frac{1}{2}(2 - 4a) = 3(2) - 5

Since

x = 2

is a solution to the equation, substituting it into the equation must yield a true mathematical statement.

2

Simplify the constant terms and distribute or simplify the expressions.

\frac{1}{3}(2a - 2) - (1 - 2a) = 1

Simplifying

3(2) - 5

yields

1

, and simplifying the second fraction term gives

\frac{1}{2}(2 - 4a) = 1 - 2a

.

3

Multiply the entire equation by 3 to eliminate the fraction.

(2a - 2) - 3(1 - 2a) = 3

Multiplying all terms on both sides of the equation by the denominator 3 clears the fraction and simplifies the equation.

4

Distribute the negative 3 and combine like terms to solve for

a

.

2a - 2 - 3 + 6a = 3 \implies 8a - 5 = 3 \implies 8a = 8 \implies a = 1

Distributing

-3(1 - 2a)

yields

-3 + 6a

. Combining like terms gives

8a - 5 = 3

, and adding 5 followed by dividing by 8 isolates the variable

a

.

Key Concept

Solving linear equations in one variable containing fractions and constant parameters by substitution and term isolation.

Estimated Time:2m 0s

Question 3Question →

If $5 - 2(x + 3) = -9$ , what is the value of $x$ ?

10

5

4

-5

Show answer & explanation

Answer: 4

Answer

4

Distributing the

-2

to both terms in the parentheses gives the equation

5 - 2x - 6 = -9

. Combining the constants on the left side yields

-2x - 1 = -9

. Adding

1

to both sides gives

-2x = -8

. Finally, dividing by

-2

yields the correct value of

4

.

Step-by-Step Solution

1

Distribute the

-2

to both terms inside the parentheses:

-2

times

x

and

-2

times

3

.

5 - 2x - 6 = -9

To remove the parentheses and simplify the equation.

2

Combine the constant terms on the left side of the equation:

5 - 6

.

-2x - 1 = -9

To group like terms together before isolating the variable.

3

Add

1

to both sides of the equation.

-2x = -8

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

4

Divide both sides of the equation by

-2

.

x = 4

To solve for

x

.

Key Concept

Solving a linear equation in one variable by distributing, combining like terms, and isolating the variable.

Question 4Question →

Two lines, $L_1$ and $L_2$ , are graphed in the $xy$ -plane. Line $L_1$ passes through the points $(2, 11)$ and $(6, 23)$ . If line $L_2$ is perpendicular to line $L_1$ and contains the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of line $L_2$ ?

Show answer & explanation

Answer: 33

Answer

33

To find the

x

-coordinate of the

x

-intercept of line

L_2

, first determine the slope of line

L_1

from the given points

(2, 11)

and

(6, 23)

using the formula

m = \frac{y_2 - y_1}{x_2 - x_1}

, which results in

m_1 = \frac{23 - 11}{6 - 2} = 3

. Since line

L_2

is perpendicular to line

L_1

, its slope must be the negative reciprocal of 3, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of line

L_2

is

y - 10 = -\frac{1}{3}(x - 3)

, which simplifies to

y = -\frac{1}{3}x + 11

. Setting

y = 0

to find the

x

-intercept yields

0 = -\frac{1}{3}x + 11

, which simplifies to

x = 33

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

using the points

(2, 11)

and

(6, 23)

with the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

m_1 = \frac{23 - 11}{6 - 2} = \frac{12}{4} = 3

The slope of a line represents its rate of change and is required to find the relationship with perpendicular lines.

2

Find the slope of line

L_2

which is perpendicular to

L_1

.

m_2 = -\frac{1}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Write the linear equation for line

L_2

using the point-slope formula

y - y_1 = m(x - x_1)

with the point

(3, 10)

and slope

m_2 = -\frac{1}{3}

.

y - 10 = -\frac{1}{3}(x - 3) \implies y = -\frac{1}{3}x + 11

Defining the equation of the line allows us to find the coordinates of any of its intercepts.

4

Set

y = 0

in the equation for line

L_2

to find the

x

-coordinate of the

x

-intercept.

0 = -\frac{1}{3}x + 11 \implies \frac{1}{3}x = 11 \implies x = 33

The

x

-intercept is the point where the line crosses the

x

-axis, which mathematically corresponds to

y = 0

.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The

x

-intercept of a linear graph represents the value of

x

when

y = 0

.

Question 5Question →

Consider the system of linear equations below.

\begin{aligned} 3x - 4y &= 7 \\ 2x + 3y &= 16 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + y$ ?

7

6

14

-9

Show answer & explanation

Answer: 7

Answer

7

The correct answer is 7. By multiplying the first equation by 3 and the second by 4, we get

9x - 12y = 21

and

8x + 12y = 64

. Adding these equations eliminates the

y

terms and gives

17x = 85

, which simplifies to

x = 5

. Substituting

x = 5

into the second equation yields

2(5) + 3y = 16 \implies 10 + 3y = 16 \implies 3y = 6 \implies y = 2

. The sum of the coordinates is

x + y = 5 + 2 = 7

.

Step-by-Step Solution

1

Multiply the equations by constants to align the coefficients of one variable for elimination.

Multiplying the first equation by 3 gives

9x - 12y = 21

. Multiplying the second equation by 4 gives

8x + 12y = 64

.

This sets up the system so that the

y

terms have opposite coefficients (

-12

and

+12

), allowing them to be eliminated by addition.

2

Add the two modified equations to eliminate the

y

variable and solve for

x

.

(9x - 12y) + (8x + 12y) = 21 + 64 \implies 17x = 85 \implies x = 5

.

Adding the equations eliminates

y

and leaves a single linear equation in terms of

x

.

3

Substitute the value of

x

back into one of the original equations to solve for

y

.

Substituting

x = 5

into the second equation

2x + 3y = 16

gives

2(5) + 3y = 16 \implies 10 + 3y = 16 \implies 3y = 6 \implies y = 2

.

Plugging the known variable value back in allows us to solve for the remaining unknown variable.

4

Calculate the value of the requested expression

x + y

.

x + y = 5 + 2 = 7

.

The question specifically asks for the sum of

x

and

y

.

Key Concept

Solving systems of linear equations using the elimination method and evaluating linear combinations of the solutions.

Alternative Method

Alternatively, you can solve the first equation for

x

in terms of

y

:

3x = 7 + 4y \implies x = \frac{7+4y}{3}

. Substitute this expression into the second equation:

2\left(\frac{7+4y}{3}\right) + 3y = 16

. Multiply the entire equation by 3 to clear the fraction:

2(7+4y) + 9y = 48 \implies 14 + 8y + 9y = 48 \implies 17y = 34 \implies y = 2

. Substitute

y = 2

back to find

x

:

x = \frac{7+4(2)}{3} = 5

. Thus,

x + y = 5 + 2 = 7

.

Estimated Time:1m 30s

Question 6Question →

A digital marketing firm allocates its monthly advertising budget between search engine campaigns and social media campaigns. Let $x$ represent the amount, in thousands of dollars, spent on search engine campaigns, and let $y$ represent the amount, in thousands of dollars, spent on social media campaigns. The system of inequalities below represents the firm's monthly constraints:

\begin{aligned} x &\ge 2y \\ x + y &\le 25 \\ 3x + 4y &\le 80 \\ y &\ge 3 \end{aligned}

Based on these constraints, what is the maximum possible amount, in thousands of dollars, the firm can spend on social media campaigns?

Show answer & explanation

Answer: 8

Answer

8

To find the maximum possible value of

y

, we analyze the boundaries of the feasible region. The boundary lines of the system are

x = 2y

,

x + y = 25

,

3x + 4y = 80

, and

y = 3

. We can find the upper limit of

y

by combining the inequalities

x \ge 2y

and

3x + 4y \le 80

. Multiplying the first inequality by

3

gives

3x \ge 6y

. Adding

4y

to both sides yields

3x + 4y \ge 10y

. Because

3x + 4y \le 80

, it follows that

10y \le 80

, which simplifies to

y \le 8

. We verify that the point

(16, 8)

satisfies the remaining inequalities:

16 + 8 = 24 \le 25

and

8 \ge 3

, which are both true. Thus, the maximum possible value of

y

is

8

.

Step-by-Step Solution

1

Relate the variables using the constraints to establish an upper bound for

y

.

Since

x \ge 2y

, multiplying both sides by

3

gives

3x \ge 6y

.

This allows us to express the

3x

term in the cost inequality in terms of

y

to determine the maximum boundary.

2

Substitute

3x \ge 6y

into the inequality

3x + 4y \le 80

.

We get

6y + 4y \le 3x + 4y \le 80

, which simplifies to

10y \le 80

.

This establishes a direct upper limit for

y

based on the intersection of the two active boundary lines.

3

Solve the inequality

10y \le 80

for

y

.

y \le 8

.

This determines that the maximum possible value for

y

under these constraints is

8

.

4

Verify that the point corresponding to

y = 8

satisfies all other constraints in the system.

When

y = 8

, the boundary

x = 2y

gives

x = 16

. Checking

(16, 8)

against all inequalities:
-

16 \ge 2(8) \implies 16 \ge 16

(True)
-

16 + 8 \le 25 \implies 24 \le 25

(True)
-

3(16) + 4(8) \le 80 \implies 80 \le 80

(True)
-

8 \ge 3

(True)

We must verify that the optimal vertex lies within the feasible region defined by all four inequalities.

Key Concept

Maximizing a coordinate value within a bounded feasible region defined by a system of linear inequalities.

Question 7Question →

A landscaping company uses the equation $12x + 18y = 240$ to model the total cost, in dollars, of renting a wood chipper for $x$ hours and a soil aerator for $y$ hours. If the wood chipper was rented for 5 more hours than the soil aerator, and the company spent a total of $240$ on these rentals, for how many hours was the soil aerator rented?

5

6

10

11

Show answer & explanation

Answer: 6

Answer

The soil aerator was rented for 6 hours.

The correct answer is 6. By writing the relationship between the rental hours as

x = y + 5

and substituting this expression into the cost equation, we obtain

12(y + 5) + 18y = 240

. Distributing the 12 results in

12y + 60 + 18y = 240

. Combining the like terms gives

30y + 60 = 240

. Subtracting 60 from both sides of the equation yields

30y = 180

. Dividing both sides by 30 results in

y = 6

. Therefore, the soil aerator was rented for 6 hours.

Step-by-Step Solution

1

Express the relationship between the rental hours of the wood chipper (

x

) and the soil aerator (

y

) as an equation.

x = y + 5

The wood chipper was rented for 5 more hours than the soil aerator.

2

Substitute the expression for

x

into the cost equation

12x + 18y = 240

.

12(y + 5) + 18y = 240

Substitution reduces the equation to a single variable, allowing us to solve for

y

.

3

Distribute the 12 and combine like terms.

12y + 60 + 18y = 240 \implies 30y + 60 = 240

Simplification isolates the variable terms on one side.

4

Isolate

y

by subtracting 60 from both sides and then dividing by 30.

30y = 180 \implies y = 6

This calculation yields the rental hours for the soil aerator.

Key Concept

Solving a system of linear equations in two variables where one equation represents a budget constraint and the other relates the quantities of the two variables.

Question 8Question →

A commercial agricultural irrigation system distributes liquid fertilizer from a storage tank. The volume of fertilizer remaining in the tank, $V$ , in liters, $m$ minutes after the system is turned on is modeled by the equation:

V = 1,800 - 15(m - 3c)

where $c$ is the number of times the system's nozzles are cleaned during the irrigation process. During each cleaning cycle, the flow of fertilizer is completely paused, and no fertilizer is distributed. Based on the model, what is the duration, in minutes, of a single cleaning cycle?

Show answer & explanation

Answer: 3

Answer

The duration of a single cleaning cycle is 3 minutes.

In the model

V = 1,800 - 15(m - 3c)

, the term

(m - 3c)

represents the total time, in minutes, that the irrigation system is actively distributing fertilizer. Since the total elapsed time is

m

minutes and the system pauses during cleaning cycles, the term

3c

represents the total paused time for

c

cleanings. Therefore, the duration of a single cleaning cycle is

\frac{3c}{c} = 3

minutes.

Step-by-Step Solution

1

Analyze the structure of the equation to identify the meaning of each term.

The coefficient 15 is the active flow rate in liters per minute, and

(m - 3c)

is the active distribution time in minutes.

To understand how the time variables affect the volume of fertilizer remaining.

2

Relate the total elapsed time to the active time and the paused time.

The total elapsed time is

m

minutes, and the active time is

m - 3c

minutes, meaning the system is paused for a total of

3c

minutes.

To find the expression for the total duration of all cleaning cycles.

3

Determine the duration of a single cleaning cycle.

Since

c

cleaning cycles result in a total pause of

3c

minutes, each cycle lasts

\frac{3c}{c} = 3

minutes.

To calculate the duration of one individual cleaning cycle.

Key Concept

Interpreting Linear Relationships in Context

Question 9Question →

In the system of equations below, what is the value of $y$ ?

4x + 3y = 25

2x + 3y = 17

Show answer & explanation

Answer: 3

Answer

The value of

y

is

3

.

Subtracting the second equation from the first equation gives

(4x + 3y) - (2x + 3y) = 25 - 17

, which simplifies to

2x = 8

. Dividing by 2 gives

x = 4

. Substituting

x = 4

into the second equation gives

2(4) + 3y = 17

, which simplifies to

8 + 3y = 17

. Subtracting 8 from both sides gives

3y = 9

, and dividing by 3 gives

y = 3

.

Step-by-Step Solution

1

Subtract the second equation from the first equation to eliminate the

y

term.

2x = 8

Subtracting the equations eliminates

3y

since it is common to both equations, leaving a single variable equation.

2

Solve for

x

by dividing both sides of the equation by 2.

x = 4

Dividing isolates the variable

x

so we can find its numerical value.

3

Substitute

x = 4

into the second equation

2x + 3y = 17

and solve for

y

.

y = 3

Substituting the value of

x

leaves only the variable

y

, which can then be isolated and solved.

Key Concept

Solving systems of linear equations using elimination

Question 10Question →

Two linear equations are defined as follows:

3x + 2y = 12

x - 2y = 4

If the ordered pair $(x, y)$ satisfies both equations, what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Adding the two equations

3x + 2y = 12

and

x - 2y = 4

eliminates the

y

terms, resulting in

4x = 16

. Dividing both sides by 4 gives the value of

x

as 4.

Step-by-Step Solution

1

Add the two equations together to eliminate the

y

variable.

4x = 16

Since the coefficients of

y

are opposites (

2

and

-2

), adding the equations eliminates

y

directly.

2

Divide both sides of the equation by 4.

x = 4

To isolate the variable

x

.

Key Concept

Solving systems of linear equations using the elimination method.

Alternative Method

We can solve the second equation for

x

to get

x = 2y + 4

. Substituting this expression into the first equation gives

3(2y + 4) + 2y = 12

, which simplifies to

6y + 12 + 2y = 12

, or

8y = 0

, meaning

y = 0

. Substituting

y = 0

back into

x = 2y + 4

yields

x = 4

.

Estimated Time:45s

Question 11Question →

If $\frac{2}{3}\left(3x - \frac{1}{2}\right) - \frac{3}{4}\left(2x - \frac{1}{3}\right) = \frac{1}{6}(x + 4) - \frac{5}{12}$ , what is the value of $12x - 5$ ?

Show answer & explanation

Answer: 7

Answer

7

The correct answer is 7. Expanding the left side of the equation yields

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

. Expanding the right side yields

\frac{1}{6}x + \frac{4}{6} - \frac{5}{12} = \frac{1}{6}x + \frac{1}{4}

. Setting the two sides equal gives

\frac{1}{2}x - \frac{1}{12} = \frac{1}{6}x + \frac{1}{4}

. Multiplying all terms by 12 clears the fractions, resulting in

6x - 1 = 2x + 3

. Solving for

x

gives

4x = 4

, which means

x = 1

. Substituting

x = 1

into

12x - 5

yields

12(1) - 5 = 7

.

Step-by-Step Solution

1

Distribute the factors on the left side of the equation:

\frac{2}{3}\left(3x - \frac{1}{2}\right)

and

-\frac{3}{4}\left(2x - \frac{1}{3}\right)

.

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

To eliminate the parentheses and prepare the left side of the equation for combining like terms.

2

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

\left(2x - \frac{3}{2}x\right) + \left(-\frac{1}{3} + \frac{1}{4}\right)

.

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

To simplify the left side into a single linear expression with a common denominator for the constants.

3

Expand and simplify the right side of the equation:

\frac{1}{6}(x + 4) - \frac{5}{12}

.

\frac{1}{6}x + \frac{1}{4}

By distributing

\frac{1}{6}

, we get

\frac{1}{6}x + \frac{4}{6} - \frac{5}{12}

. Finding a common denominator of 12 for the constant terms yields

\frac{8}{12} - \frac{5}{12} = \frac{3}{12} = \frac{1}{4}

.

4

Equate the simplified left and right sides:

\frac{1}{2}x - \frac{1}{12} = \frac{1}{6}x + \frac{1}{4}

. Multiply both sides of the equation by 12 to clear the fractions.

6x - 1 = 2x + 3

, which simplifies to

4x = 4

, and thus

x = 1

.

To solve for the variable

x

in a simplified integer form.

5

Substitute

x = 1

into the requested expression

12x - 5

.

12(1) - 5 = 7

To calculate the final value asked by the question.

Key Concept

Solving multi-step linear equations in one variable with fractional coefficients, distributing negative signs, and evaluating expressions.

Question 12Question →

The graph of the linear equation $y = 3x - 7$ contains the point $(k, 5)$ . What is the value of $k$ ?

-\frac{2}{3}

4

8

-4

Show answer & explanation

Answer: $4$

Answer

The value of

k

is

4

.

Substituting the coordinates of the point

(k, 5)

into the linear equation

y = 3x - 7

gives

5 = 3k - 7

. Solving for

k

involves adding

7

to both sides of the equation to get

12 = 3k

, and then dividing both sides by

3

to find

k = 4

.

Step-by-Step Solution

1

Substitute the point

(k, 5)

into the equation of the line

y = 3x - 7

by setting

x = k

and

y = 5

.

5 = 3k - 7

Since the point lies on the graph of the equation, its coordinates must satisfy the equation.

2

Isolate the term with

k

by adding

7

to both sides of the equation.

12 = 3k

Adding

7

to both sides cancels the

-7

on the right side.

3

Solve for

k

by dividing both sides of the equation by

3

.

k = 4

Dividing by

3

isolates

k

.

Key Concept

Determining an unknown coordinate of a point on a line by substituting the point's coordinates into the linear equation in two variables.

Question 13Question →

Let $p$ and $q$ be constants. The inequality $p(x - 5) < q(x + 4)$ has the solution set $x > 2$ . If $p - q = -3$ , what is the value of $p + q$ ?

1

39

-1

-3

Show answer & explanation

Answer: -1

Answer

-1

The correct answer is

-1

. Expanding the inequality

p(x - 5) < q(x + 4)

gives

px - 5p < qx + 4q

. Grouping the

x

terms yields

(p - q)x < 5p + 4q

. Since

p - q = -3

, which is negative, dividing both sides by

p - q

reverses the inequality to

x > \frac{5p + 4q}{p - q}

. Comparing this to the given solution

x > 2

, the boundary value must satisfy

\frac{5p + 4q}{-3} = 2

, which simplifies to

5p + 4q = -6

. Using the system of equations

p - q = -3

and

5p + 4q = -6

, we find

p = -2

and

q = 1

. Therefore, the value of

p + q

is

-2 + 1 = -1

.

Step-by-Step Solution

1

Expand both sides of the inequality and group the terms with

x

on one side.

(p - q)x < 5p + 4q

Expanding

p(x - 5) < q(x + 4)

gives

px - 5p < qx + 4q

. Subtracting

qx

and adding

5p

to both sides yields

px - qx < 5p + 4q

, which factors to

(p - q)x < 5p + 4q

.

2

Determine the direction of the inequality when isolating

x

.

x > \frac{5p + 4q}{p - q}

Since we are given

p - q = -3

, the quantity

p - q

is negative. Dividing both sides of the inequality by a negative value reverses the inequality symbol from

<

to

>

.

3

Equate the resulting boundary expression to the boundary of the given solution set

x > 2

.

5p + 4q = -6

The boundary value of the solution set is

2

, so we set

\frac{5p + 4q}{p - q} = 2

. Substituting

p - q = -3

gives

\frac{5p + 4q}{-3} = 2

, which simplifies to

5p + 4q = -6

.

4

Solve the system of linear equations for

p

and

q

.

p = -2

and

q = 1

We have the system of equations

p - q = -3

and

5p + 4q = -6

. From the first equation,

p = q - 3

. Substituting this into the second equation gives

5(q - 3) + 4q = -6 \implies 9q - 15 = -6 \implies 9q = 9 \implies q = 1

. Substituting

q = 1

back into

p = q - 3

gives

p = -2

.

5

Calculate the value of

p + q

.

-1

Adding the values of

p

and

q

gives

p + q = -2 + 1 = -1

.

Key Concept

Solving linear inequalities in one variable with variable coefficients and applying sign reversal rules.

Question 14Question →

A community garden charges a one-time registration fee plus a monthly fee to plot and maintain a garden bed. The total cost, $y$ , in dollars, of maintaining a garden bed for $x$ months is given by the equation $y = 15x + 45$ . If a gardener spent a total of $165$ dollars, for how many months did they maintain the garden bed?

Show answer & explanation

Answer: 8

Answer

The correct answer is 8. Setting the total cost y equal to 165 in the linear equation yields 165 = 15x + 45. Subtracting 45 from both sides gives 120 = 15x, and dividing by 15 results in x = 8 months.

To find the number of months the gardener maintained the garden bed, substitute the total spent,

165

, for

y

in the equation

y = 15x + 45

. This results in

165 = 15x + 45

. Subtracting

45

from both sides of the equation yields

120 = 15x

. Dividing both sides by

15

gives

x = 8

. Thus, the garden bed was maintained for 8 months.

Step-by-Step Solution

1

Substitute 165 for y in the equation.

165 = 15x + 45

The variable y represents the total cost in dollars, which is given as 165.

2

Subtract 45 from both sides of the equation.

120 = 15x

This isolates the variable term by subtracting the constant registration fee.

3

Divide both sides of the equation by 15.

x = 8

This solves for x, the number of months.

Key Concept

Solving for a variable in a linear equation representing a real-world scenario.

Question 15Question →

An online bookstore charges a flat shipping fee plus a fixed price per book purchased. The total cost, $y$ , in dollars, for purchasing $x$ books is given by the equation $y = 8.5x + 4.5$ . If a customer's total cost was $47$ dollars, how many books did the customer purchase?

Show answer & explanation

Answer: 5

Answer

The customer purchased 5 books.

To find the number of books purchased, substitute the total cost of 47 for

y

in the equation

y = 8.5x + 4.5

, yielding

47 = 8.5x + 4.5

. Subtracting

4.5

from both sides gives

42.5 = 8.5x

. Dividing both sides by

8.5

results in

x = 5

. Thus, the customer purchased 5 books.

Step-by-Step Solution

1

Substitute the total cost of 47 into the equation for y

47 = 8.5x + 4.5

Since the total cost is represented by y and is given as 47 dollars, we substitute 47 for y in the equation.

2

Subtract 4.5 from both sides of the equation

42.5 = 8.5x

Subtracting the flat fee from both sides isolates the variable term representing the total cost of the books.

3

Divide both sides by 8.5

x = 5

Dividing the remaining cost by the price per book yields the total number of books purchased.

Key Concept

Linear Equations in Two Variables

Question 16Question →

If $\frac{1}{2}(4x - 8) - \frac{2}{3}(3 - 9x) = 5(x - 1) + 7$ , what is the value of $3x - 2$ ?

6

2

10

-\frac{14}{3}

Show answer & explanation

Answer: 6

Answer

6

The correct value is 6. After correctly distributing the coefficients on both sides, the equation becomes

8x - 6 = 5x + 2

. Isolating the variable yields

3x = 8

, which gives

x = \frac{8}{3}

. Substituting this back into the target expression

3x - 2

results in

3(\frac{8}{3}) - 2 = 6

.

Step-by-Step Solution

1

Distribute the fraction coefficients on the left side of the equation.

\frac{1}{2}(4x - 8) - \frac{2}{3}(3 - 9x) = 2x - 4 - 2 + 6x = 8x - 6

To simplify the expression by removing the parentheses on the left side.

2

Distribute and simplify the right side of the equation.

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

To simplify the expression by removing the parentheses on the right side.

3

Equate the simplified left and right sides, then solve for

x

.

8x - 6 = 5x + 2 \implies 3x = 8 \implies x = \frac{8}{3}

To isolate the variable

x

.

4

Substitute the value of

x

into the expression

3x - 2

.

3\left(\frac{8}{3}\right) - 2 = 8 - 2 = 6

To find the final value requested by the question.

Key Concept

Solving linear equations in one variable by distributing coefficients, combining like terms, and isolating the variable.

Estimated Time:2m 0s

Question 17Question →

In the equation below,

x

is a real number.

\frac{5}{6}(3x - 4) - \frac{3}{8}(4x - 12) = \frac{1}{4}(2x + 6)

What is the value of

6x - 5

?

107

-1

7

-21

Show answer & explanation

Answer: $-1$

Answer

-1

The correct answer is

-1

. First, distribute the fractions across the parentheses on both sides of the equation:

\frac{5}{6}(3x - 4) - \frac{3}{8}(4x - 12) = \frac{1}{4}(2x + 6)

\frac{5}{2}x - \frac{10}{3} - \frac{3}{2}x + \frac{9}{2} = \frac{1}{2}x + \frac{3}{2}

Combine the like terms on the left side:

\left(\frac{5}{2}x - \frac{3}{2}x\right) + \left(-\frac{10}{3} + \frac{9}{2}\right) = \frac{1}{2}x + \frac{3}{2}

x + \frac{7}{6} = \frac{1}{2}x + \frac{3}{2}

Subtract

\frac{1}{2}x

from both sides:

\frac{1}{2}x + \frac{7}{6} = \frac{3}{2}

Subtract

\frac{7}{6}

from both sides:

\frac{1}{2}x = \frac{9}{6} - \frac{7}{6} = \frac{2}{6} = \frac{1}{3}

Multiply by

2

to solve for

x

:

x = \frac{2}{3}

Finally, substitute

x = \frac{2}{3}

into the expression

6x - 5

:

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

Step-by-Step Solution

1

Distribute the coefficients to the terms within the parentheses on both sides of the equation.

\frac{5}{2}x - \frac{10}{3} - \frac{3}{2}x + \frac{9}{2} = \frac{1}{2}x + \frac{3}{2}

To simplify the linear equation, parenthetical expressions must be expanded.

2

Combine like terms on the left side of the equation.

x + \frac{7}{6} = \frac{1}{2}x + \frac{3}{2}

Grouping the variable terms (

x

) and constant terms simplifies the equation prior to isolation.

3

Isolate the variable term on one side of the equation by subtracting

\frac{1}{2}x

and

\frac{7}{6}

from both sides.

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

This isolates the variable

x

on the left side and the constants on the right side.

4

Solve for

x

by multiplying both sides by

2

.

x = \frac{2}{3}

Multiplying by the reciprocal of the coefficient of

x

gives the value of

x

.

5

Substitute the value of

x

into the expression

6x - 5

.

-1

The question asks for the value of the expression

6x - 5

, not the variable

x

.

Key Concept

Solving multi-step linear equations in one variable containing fractions and parentheses.

Question 18Question →

If $5(x - 2) = 3x + 8$ , what is the value of $x$ ?

5

6

9

-1

Show answer & explanation

Answer: 9

Answer

The value of

x

is 9.

Distributing the 5 to both terms inside the parentheses yields

5x - 10 = 3x + 8

. Subtracting

3x

from both sides gives

2x - 10 = 8

. Adding 10 to both sides yields

2x = 18

. Dividing both sides by 2 gives the correct value of

x

, which is 9.

Step-by-Step Solution

1

Distribute the 5 to the terms inside the parentheses on the left side of the equation

5x - 10 = 3x + 8

To simplify the left side of the equation by removing the parentheses.

2

Subtract

3x

from both sides of the equation to collect the variable terms on one side

2x - 10 = 8

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

3

Add 10 to both sides of the equation to collect the constant terms on the other side

2x = 18

To further isolate the variable term.

4

Divide both sides of the equation by 2

x = 9

To solve for the variable

x

.

Key Concept

Solving linear equations in one variable using distributive property and isolation of the variable.

Question 19Question →

If $x$ is the solution to the equation $\frac{1}{4}\left(\frac{2}{3}x - 8\right) - \frac{5}{6}\left(12 - \frac{3}{5}x\right) = \frac{1}{3}(x - 9) - 1$ , what is the value of $\frac{1}{2}x - 5$ ?

7

-11

1

-29

Show answer & explanation

Answer: 7

Answer

7

The correct answer is obtained by first distributing the fractions across the terms in the parentheses, which yields

\frac{1}{6}x - 2 - 10 + \frac{1}{2}x = \frac{1}{3}x - 4

. Combining the like terms on the left side gives

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

. Subtracting

\frac{1}{3}x

from both sides and adding

12

to both sides yields

\frac{1}{3}x = 8

, which means

x = 24

. Substituting

24

for

x

in the expression

\frac{1}{2}x - 5

gives

\frac{1}{2}(24) - 5 = 12 - 5 = 7

.

Step-by-Step Solution

1

Distribute the coefficients outside the parentheses on both sides of the equation.

\frac{1}{6}x - 2 - 10 + \frac{1}{2}x = \frac{1}{3}x - 3 - 1

To eliminate parentheses and simplify the terms.

2

Combine the like terms on the left and right sides of the equation.

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

To group the coefficients of

x

and the constant values.

3

Isolate the variable

x

by subtracting

\frac{1}{3}x

and adding

12

to both sides of the equation.

\frac{1}{3}x = 8 \implies x = 24

To solve for

x

.

4

Substitute the value of

x

into the expression

\frac{1}{2}x - 5

to find the final value.

\frac{1}{2}(24) - 5 = 12 - 5 = 7

To evaluate the requested expression.

Key Concept

Solving multi-step linear equations in one variable involving distribution, fractions, and grouping like terms.

Alternative Method

Instead of distributing fractions first, we can multiply the entire equation by the least common multiple of all denominators, which is

12

. This eliminates all fractions immediately:

3\left(\frac{2}{3}x - 8\right) - 10\left(12 - \frac{3}{5}x\right) = 4(x - 9) - 12

. Expanding this gives

2x - 24 - 120 + 6x = 4x - 36 - 12 \implies 8x - 144 = 4x - 48 \implies 4x = 96 \implies x = 24

. Substituting

x = 24

into the expression gives

7

.

Estimated Time:3m 0s

Question 20Question →

If $2(3x - 4) - 5(x - 1) = 3(x + 2) - 1$ , what is the value of $2x + 3$ ?

-15

-5

5

11

Show answer & explanation

Answer: -5

Answer

The correct answer is

-5

.

Solving the given equation step-by-step yields

x = -4

. Substituting

x = -4

into the expression

2x + 3

results in

2(-4) + 3 = -5

.

Step-by-Step Solution

1

Distribute the values outside the parentheses on both sides of the equation.

6x - 8 - 5x + 5 = 3x + 6 - 1

This step removes parentheses so that like terms can be combined.

2

Combine like terms on each side of the equation.

x - 3 = 3x + 5

Simplifying both sides makes it easier to isolate the variable.

3

Isolate the variable

x

by subtracting

x

and

5

from both sides.

x = -4

This determines the value of

x

which is required to evaluate the final expression.

4

Substitute the value of

x

into the expression

2x + 3

.

2(-4) + 3 = -5

This calculates the final value requested by the question.

Key Concept

Solving linear equations in one variable involving parentheses and distributing terms.

Algebra

Answer

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Answer

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Alternative Method