Algebra Practice Questions — SAT — Page 13

The table below shows several values of $x$ and the corresponding values of two linear functions, $f$ and $g$ .

$x$	$f(x)$	$g(x)$
$-2$	$12$	$-3$
$0$	$6$	$1$
$2$	$0$	$5$
$4$	$-6$	$9$

If the system of equations $y = f(x)$ and $y = g(x)$ has solution $(x, y)$ , what is the value of $x + y$ ?

-20

4

10

\frac{16}{7}

Show answer & explanation

Answer: $4$

Answer

The sum of the coordinates of the solution to the system is

4

.

To find the solution to the system

y = f(x)

and

y = g(x)

, we must first determine the equations of the linear functions

f

and

g

from the given table. For

f(x)

, using the points

(0, 6)

and

(2, 0)

, the slope is

\frac{0 - 6}{2 - 0} = -3

. Since the

y

-intercept is

6

, the equation is

f(x) = -3x + 6

. For

g(x)

, using the points

(0, 1)

and

(2, 5)

, the slope is

\frac{5 - 1}{2 - 0} = 2

. Since the

y

-intercept is

1

, the equation is

g(x) = 2x + 1

. Setting the two equations equal to find their intersection gives

-3x + 6 = 2x + 1

. Solving for

x

yields

5x = 5

, or

x = 1

. Substituting

x = 1

into

g(x)

gives

y = 2(1) + 1 = 3

. The sum of the coordinates of the solution is

x + y = 1 + 3 = 4

.

Step-by-Step Solution

1

Determine the linear equation for

f(x)

using the table values.

f(x) = -3x + 6

The slope of

f

is calculated as

\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 6}{2} = -3

. Since

f(0) = 6

, the

y

-intercept is

6

.

2

Determine the linear equation for

g(x)

using the table values.

g(x) = 2x + 1

The slope of

g

is calculated as

\frac{g(2) - g(0)}{2 - 0} = \frac{5 - 1}{2} = 2

. Since

g(0) = 1

, the

y

-intercept is

1

.

3

Set

f(x) = g(x)

to solve for the

x

-coordinate of the intersection.

x = 1

Equating the two expressions gives

-3x + 6 = 2x + 1

. Adding

3x

to both sides and subtracting

1

from both sides results in

5x = 5

, which simplifies to

x = 1

.

4

Substitute

x = 1

back into either equation to solve for

y

.

y = 3

Using the equation

g(x) = 2x + 1

, substituting

x = 1

gives

y = 2(1) + 1 = 3

.

5

Calculate the sum

x + y

.

4

Adding the coordinates of the solution

(1, 3)

gives

1 + 3 = 4

.

Key Concept

Solving a system of linear equations derived from a table of values.

Question 242Question →

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

15x - 2(3x - k) = 9x + 10

If the equation has infinitely many solutions, what is the value of $k$ ?

-5

20

5

8

Show answer & explanation

Answer: 5

Answer

The correct value of

k

is

5

, which makes the equation have infinitely many solutions.

The correct answer is

5

because distributing

-2

through the parentheses on the left side of the equation yields

15x - 6x + 2k = 9x + 10

. Combining the

x

terms gives

9x + 2k = 9x + 10

. For a linear equation in one variable to have infinitely many solutions, both sides must be identical. Since the coefficients of

x

are already equal (

9 = 9

), we set the constants equal to each other (

2k = 10

), which simplifies to

k = 5

.

Step-by-Step Solution

1

Distribute the constant

-2

to both terms inside the parentheses on the left side of the equation.

15x - 6x + 2k = 9x + 10

Applying the distributive property correctly simplifies the terms inside the parentheses.

2

Combine the like terms of

x

on the left side of the equation.

9x + 2k = 9x + 10

Subtracting

6x

from

15x

simplifies the left side of the equation to have a single

x

term.

3

Set the constant terms on both sides equal to each other to find the condition for infinitely many solutions.

2k = 10

For a linear equation in one variable to have infinitely many solutions, the coefficients of the variable on both sides must be equal, and the constant terms on both sides must also be equal.

4

Solve for

k

by dividing both sides of the equation by

2

.

k = 5

Isolating the variable

k

yields the final value.

Key Concept

For a linear equation in one variable to have infinitely many solutions, the equation must simplify to an identity where the variable coefficients are equal and the constant terms are equal on both sides of the equation.

Estimated Time:1m 15s

Question 243Question →

A logistics company operates two distribution centers, Center A and Center B. The daily operating cost, $C_A$ , in dollars, at Center A when processing $p$ packages is given by the equation $C_A = 1.75p + 3,200$ . The daily operating cost, $C_B$ , in dollars, at Center B when processing $p$ packages is given by the equation $C_B = 2.25(p - 400) + 2,800$ , where $p \geq 400$ . The difference in daily operating costs, $D$ , in dollars, between Center B and Center A is defined as $D = C_B - C_A$ . For $p \geq 400$ , this difference is modeled by the equation $D = 0.50p - 1,300$ . Which of the following is the best interpretation of the number $0.50$ in this context?

For each additional package processed, the daily operating cost at Center A increases by

0.50

dollars more than the daily operating cost at Center B.

For each additional package processed, the daily operating cost at Center B increases by

0.50

dollars more than the daily operating cost at Center A.

For each increase of

2.00

dollars in the difference of daily operating costs between the two centers, Center B processes

1

additional package.

The cost to process each package at Center B is

0.50

dollars.

Show answer & explanation

Answer: For each additional package processed, the daily operating cost at Center B increases by $0.50$ dollars more than the daily operating cost at Center A.

Answer

For each additional package processed, the daily operating cost at Center B increases by

0.50

dollars more than the daily operating cost at Center A.

The equation

D = 0.50p - 1,300

models the difference in daily operating costs,

D

, between Center B and Center A (

C_B - C_A

) as a function of the number of packages processed,

p

. The coefficient of

p

, which is

0.50

, represents the slope of this linear relationship. This means that for each unit increase in

p

(each additional package processed), the difference

D

increases by

0.50

dollars. Since

D = C_B - C_A

, an increase in

D

indicates that Center B's daily operating cost is increasing by

0.50

dollars more than Center A's daily operating cost for each additional package processed.

Step-by-Step Solution

1

Understand the meaning of the variables and the difference equation

D = C_B - C_A

.

The variable

p

represents the number of packages processed, and

D

represents the difference in daily operating costs between Center B and Center A.

Establishing the relationship between the independent variable and the dependent variable is necessary to interpret the slope.

2

Identify the slope in the linear equation

D = 0.50p - 1,300

.

The equation is in slope-intercept form,

y = mx + b

, where the slope

m

is

0.50

.

The slope represents the rate of change of the dependent variable

D

with respect to the independent variable

p

.

3

Interpret the rate of change in the context of the problem.

A slope of

0.50

means that for every increase of

1

in

p

(each additional package processed), the value of

D

increases by

0.50

dollars. Since

D = C_B - C_A

, an increase of

0.50

in

D

means that

C_B

(Center B's cost) increases by

0.50

dollars more than

C_A

(Center A's cost).

This links the mathematical rate of change directly to the real-world difference in costs between the two centers.

Key Concept

Interpreting the slope of a combined linear relationship in context

Estimated Time:1m 30s

Question 244Question →

The table below shows several values of $x$ and their corresponding values of $y$ for a linear relationship, where $p$ and $k$ are constants.

$x$	$y$
$p$	$4$
$p + 2$	$10$
$2p + 3$	$k$

If $k = 25$ , what is the value of $p$ ?

0.5

4

6

60

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. A linear relationship has a constant rate of change (slope). Calculating the slope

m

from the first two points

(p, 4)

and

(p + 2, 10)

gives

m = \frac{10 - 4}{(p + 2) - p} = 3

. The equation of the line is

y - 4 = 3(x - p)

, which simplifies to

y = 3x - 3p + 4

. Substituting the third point

(2p + 3, 25)

into this equation gives

25 = 3(2p + 3) - 3p + 4

, which simplifies to

25 = 3p + 13

. Solving for

p

yields

p = 4

.

Step-by-Step Solution

1

Determine the slope of the linear relationship using the first two points from the table,

(p, 4)

and

(p + 2, 10)

.

The slope is

m = 3

.

A linear relationship has a constant rate of change (slope), which is the change in

y

divided by the change in

x

:

m = \frac{10 - 4}{(p + 2) - p} = \frac{6}{2} = 3

.

2

Write the equation of the line using the point-slope form with the point

(p, 4)

and slope

m = 3

.

y = 3x - 3p + 4

Expressing the linear relationship as an equation allows us to find the relationship between the parameters

p

and

k

:

y - 4 = 3(x - p) \Rightarrow y = 3x - 3p + 4

.

3

Substitute the third point

(2p + 3, k)

into the linear equation and simplify the expression to solve for

k

in terms of

p

.

k = 3p + 13

Since the point

(2p + 3, k)

lies on the line, its coordinates must satisfy the line's equation:

k = 3(2p + 3) - 3p + 4 \Rightarrow k = 6p + 9 - 3p + 4 \Rightarrow k = 3p + 13

.

4

Substitute the given value

k = 25

into the equation

k = 3p + 13

and solve for

p

.

p = 4

This isolates the variable

p

to find its numerical value under the specified conditions:

25 = 3p + 13 \Rightarrow 12 = 3p \Rightarrow p = 4

.

Key Concept

Finding the equation of a line from a table of values and using substitution to solve for unknown parameters in a linear relationship.

Alternative Method

An alternative method is to use the property that the slope between any two points on a line is constant. We can equate the slope between

(p, 4)

and

(p + 2, 10)

to the slope between

(p, 4)

and

(2p + 3, 25)

. This gives the equation

\frac{10 - 4}{(p + 2) - p} = \frac{25 - 4}{(2p + 3) - p}

, which simplifies to

3 = \frac{21}{p + 3}

. Solving this equation yields

3(p + 3) = 21

, so

p + 3 = 7

, which gives

p = 4

.

Estimated Time:2m 0s

Question 245Question →

An electric vehicle's battery is being charged. The charge of the battery, $C$ , as a percentage of its full capacity, $t$ minutes after the charging begins is modeled by the equation $C = 1.2t + 18$ , where $t \le 60$ . According to the model, what was the battery's charge percentage when the charging began?

Show answer & explanation

Answer: 18

Answer

The battery's charge percentage when the charging began was 18.

The linear relationship is given by the equation

C = 1.2t + 18

. The constant term in this linear model, 18, represents the y-intercept, which is the value of

C

when

t = 0

. In the context of this scenario,

t = 0

represents the time when the charging began, and

C

represents the charge percentage. Substituting

t = 0

into the equation yields

C = 1.2(0) + 18 = 18

. Therefore, the battery's charge percentage was 18 when the charging began.

Step-by-Step Solution

1

Determine the value of the independent variable

t

when charging began.

t = 0

The initial state or the start of the charging process corresponds to a time of 0 minutes.

2

Substitute

t = 0

into the equation

C = 1.2t + 18

.

C = 18

Evaluating the linear equation at

t = 0

yields the constant term, representing the initial charge percentage.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 246Question →

The table below shows some values of $x$ and the corresponding values of the linear function $f$ .

$x$	$f(x)$
$-2$	$13$
$2$	$5$
$6$	$-3$

If the graph of a second linear function, $g$ , has a slope that is twice the slope of the graph of $f$ and passes through the point $(1, 4)$ , what is the value of $g(5)$ ?

0

-20

-12

-16

Show answer & explanation

Answer: -12

Answer

-12

The correct answer is

-12

. First, calculate the slope of the linear function

f

using two coordinate pairs from the table, such as

(-2, 13)

and

(2, 5)

. The slope is

\frac{5 - 13}{2 - (-2)} = -2

. Since the slope of function

g

is twice the slope of

f

, the slope of

g

is

2 \times (-2) = -4

. Using the point-slope equation with the point

(1, 4)

, we have

g(x) - 4 = -4(x - 1)

, which simplifies to

g(x) = -4x + 8

. Finally, evaluating the function at

x = 5

gives

g(5) = -4(5) + 8 = -12

.

Step-by-Step Solution

1

Calculate the slope of the linear function

f

using two points from the table,

(-2, 13)

and

(2, 5)

.

The slope of

f

is

m_f = \frac{5 - 13}{2 - (-2)} = \frac{-8}{4} = -2

.

Finding the slope of the reference function is necessary to determine the slope of the second function.

2

Determine the slope of the function

g

.

The slope of

g

is

m_g = 2 \times (-2) = -4

.

The problem states that the slope of

g

is twice the slope of

f

.

3

Find the equation of

g(x)

using the point-slope form with the point

(1, 4)

.

g(x) - 4 = -4(x - 1) \implies g(x) = -4x + 8

.

Using a point on the line and its slope allows us to construct the full function definition.

4

Evaluate

g(5)

by substituting

x = 5

into the equation for

g(x)

.

g(5) = -4(5) + 8 = -12

.

This yields the final value requested by the question.

Key Concept

Linear Functions and Graphs

Question 247Question →

Consider the system of equations below.

\begin{aligned} 2(x + 3y) - 5y &= 10 \\ 3(x - y) + 2x &= 3 \end{aligned}

If

(x, y)

is the solution to the system, what is the value of

x + 2y

?

9

10

11

17

Show answer & explanation

Answer: 11

Answer

The value of 11 is the correct answer.

The correct answer is 11. Simplifying the first equation gives

2x + y = 10

, which allows us to write

y = 10 - 2x

. The second equation simplifies to

5x - 3y = 3

. Substituting the expression for

y

into the second equation yields

5x - 3(10 - 2x) = 3

. Distributing the

-3

gives

5x - 30 + 6x = 3

, which simplifies to

11x = 33

, or

x = 3

. Substituting

x = 3

back into the equation for

y

gives

y = 10 - 2(3) = 4

. Evaluating the expression

x + 2y

with these values results in

3 + 2(4) = 11

.

Step-by-Step Solution

1

Simplify both equations in the system by distributing coefficients and combining like terms.

The first equation becomes

2x + 6y - 5y = 10 \Rightarrow 2x + y = 10

.
The second equation becomes

3x - 3y + 2x = 3 \Rightarrow 5x - 3y = 3

.

Simplifying the equations makes it easier to use substitution or elimination methods.

2

Solve the simplified system using substitution.

From

2x + y = 10

, express

y

as

y = 10 - 2x

. Substitute this into the second equation:

5x - 3(10 - 2x) = 3 \Rightarrow 5x - 30 + 6x = 3 \Rightarrow 11x = 33 \Rightarrow x = 3

. Substituting

x = 3

back into the expression for

y

gives

y = 10 - 2(3) = 4

.

This determines the unique values of the variables

x

and

y

that satisfy both equations.

3

Evaluate the expression

x + 2y

using the solved values.

Substitute

x = 3

and

y = 4

into

x + 2y

to get

3 + 2(4) = 11

.

This finds the specific quantity requested by the question.

Key Concept

Solving systems of linear equations using algebraic simplification and substitution.

Question 248Question →

A customer opens a savings account with an initial deposit of $\$180$ and deposits $\$30$ at the end of each week. Two weeks later, a second customer opens a savings account with an initial deposit of $\$120$ and deposits $\$50$ at the end of each week. If neither customer makes any other deposits or withdrawals, after how many weeks from the time the first customer opened their account will both accounts have the same balance?

Show answer & explanation

Answer: 8

Answer

Both accounts will have the same balance after 8 weeks.

The correct answer is 8. Letting

w

represent the number of weeks since the first customer opened their account, the balance of the first account can be modeled by the expression

180 + 30w

. Since the second customer opens their account two weeks later, the number of weeks they have been depositing money is

w - 2

. Therefore, the balance of the second account can be modeled by the expression

120 + 50(w - 2)

. Setting the two expressions equal to find when the balances are the same gives the equation

180 + 30w = 120 + 50(w - 2)

. Distributing the

50

yields

180 + 30w = 120 + 50w - 100

. Simplifying the right side gives

180 + 30w = 50w + 20

. Subtracting

30w

and

20

from both sides of the equation yields

160 = 20w

. Dividing both sides by

20

gives

w = 8

.

Step-by-Step Solution

1

Set up expressions for the balance of each account after

w

weeks.

First account balance:

180 + 30w

; Second account balance:

120 + 50(w - 2)

The first customer has been depositing for

w

weeks. The second customer started 2 weeks later, so they have been depositing for

w - 2

weeks.

2

Set the two expressions equal to each other to find when the balances are equal.

180 + 30w = 120 + 50(w - 2)

We want to find the number of weeks

w

when the two account balances are equal.

3

Solve the linear equation for

w

.

w = 8

Distribute the 50 to get

180 + 30w = 120 + 50w - 100

. Combine constants to get

180 + 30w = 50w + 20

. Subtract

30w

and

20

from both sides to get

160 = 20w

. Divide by

20

to get

w = 8

.

Key Concept

Formulating and solving linear equations in one variable from real-world contexts

Question 249Question →

A delivery driver is loading a van with small boxes and large boxes. Each small box weighs $10$ pounds, and each large box weighs $35$ pounds. The total weight of the boxes in the van cannot exceed $1,000$ pounds. The driver must load at least $4$ times as many small boxes as large boxes. If the driver loads at least $10$ large boxes, what is the maximum number of large boxes that the driver can load?

Show answer & explanation

Answer: 13

Answer

13

To find the maximum number of large boxes,

y

, we set up the system of inequalities:

10x + 35y \le 1000

and

x \ge 4y

. To maximize

y

, we minimize

x

by setting

x = 4y

. Substituting this into the weight constraint gives

10(4y) + 35y \le 1000

, which simplifies to

75y \le 1000

, or

y \le 13.33

. The largest integer satisfying this inequality is 13, which also satisfies

y \ge 10

.

Step-by-Step Solution

1

Define variables for the quantities of each type of box.

Let

x

be the number of small boxes and

y

be the number of large boxes, where

x

and

y

are non-negative integers.

Establishing variables is necessary to translate the verbal constraints into algebraic inequalities.

2

Translate the given constraints into a system of linear inequalities.

The weight limit gives

10x + 35y \le 1000

. The requirement for at least 4 times as many small boxes as large boxes gives

x \ge 4y

. The minimum of 10 large boxes gives

y \ge 10

.

Representing the scenario mathematically allows us to find the boundaries of the solution set.

3

Express the weight inequality in terms of a single variable by using the boundary condition of the second inequality.

To maximize

y

, we want to minimize the weight contributed by the small boxes,

x

. The minimum value of

x

for any

y

is

x = 4y

. Substituting

x = 4y

into the weight inequality gives

10(4y) + 35y \le 1000

.

Finding the extreme case (minimum number of small boxes) provides the upper limit for the number of large boxes.

4

Simplify the inequality and solve for

y

.

40y + 35y \le 1000 \implies 75y \le 1000 \implies y \le \frac{1000}{75} \approx 13.33

.

This calculation determines the upper algebraic bound for the number of large boxes.

5

Determine the maximum integer value for

y

that satisfies the system.

Since the number of boxes must be an integer, the maximum integer value less than or equal to

13.33

is

13

. Since

13 \ge 10

, this satisfies all constraints.

Real-world quantities like boxes must be whole numbers, so we take the largest integer within the solution set.

Key Concept

Solving systems of linear inequalities in two variables to optimize a value under constraints.

Question 250Question →

A distributor plans to mix two coffee blends, Blend A and Blend B, to create a custom mixture. The table below shows the distribution of Colombian and Ethiopian coffee beans in each blend by weight:

Coffee Blend	Colombian Beans	Ethiopian Beans
Blend A	60%	40%
Blend B	20%	80%

The distributor wants the final custom mixture to contain exactly 14 kilograms of Colombian coffee beans and 16 kilograms of Ethiopian coffee beans. How many kilograms of Blend A should the distributor use to create this mixture?

Show answer & explanation

Answer: 20

Answer

The distributor should use 20 kilograms of Blend A.

By setting up a system of linear equations based on the percentage of each bean type in Blend A (

A

) and Blend B (

B

), we get

0.60A + 0.20B = 14

and

0.40A + 0.80B = 16

. Solving this system yields

A = 20

and

B = 10

. Therefore, 20 kilograms of Blend A are required.

Step-by-Step Solution

1

Define variables for the unknowns and write the system of equations.

Let

A

be the number of kilograms of Blend A, and let

B

be the number of kilograms of Blend B. The system of equations is:
For Colombian beans:

0.60A + 0.20B = 14

For Ethiopian beans:

0.40A + 0.80B = 16

This translates the verbal and tabular constraints of the problem into mathematical equations.

2

Simplify the system by multiplying both sides of each equation to eliminate decimals.

Multiply the first equation by 5:

3A + B = 70 \implies B = 70 - 3A

Multiply the second equation by 5:

2A + 4B = 80 \implies A + 2B = 40

Working with integers makes the algebraic manipulation easier and reduces arithmetic errors.

3

Substitute the expression for

B

into the simplified second equation to solve for

A

.

A + 2(70 - 3A) = 40

A + 140 - 6A = 40

-5A = -100

A = 20

This isolates the variable

A

, which represents the required kilograms of Blend A.

Key Concept

Systems of Linear Equations

Question 251Question →

For a constant $k$ , the inequality $\frac{k - 3x}{2} > \frac{5x + 3}{4}$ has exactly 4 positive integer solutions for $x$ . If $k$ is an integer, how many possible values of $k$ are there?

Show answer & explanation

Answer: 6

Answer

The number of possible integer values for

k

is 6.

Simplifying the inequality gives

x < \frac{2k - 3}{11}

. For this inequality to have exactly 4 positive integer solutions, the solution set must contain only the integers 1, 2, 3, and 4. This requires the boundary

\frac{2k - 3}{11}

to satisfy

4 < \frac{2k - 3}{11} \le 5

. Solving for

k

gives

23.5 < k \le 29

. The integers in this interval are 24, 25, 26, 27, 28, and 29, which total 6 values.

Step-by-Step Solution

1

Multiply both sides of the inequality by 4.

2(k - 3x) > 5x + 3

Clearing the denominators makes the inequality easier to solve.

2

Distribute the 2 on the left side.

2k - 6x > 5x + 3

Expanding terms allows us to group variables.

3

Add

6x

to both sides and subtract 3 from both sides.

11x < 2k - 3

Grouping

x

on one side and parameter terms on the other side.

4

Divide by 11.

x < \frac{2k - 3}{11}

Isolating

x

gives the boundary for the solution set.

5

Establish the range for the boundary term

\frac{2k - 3}{11}

.

4 < \frac{2k - 3}{11} \le 5

For the inequality to contain exactly the positive integers 1, 2, 3, and 4, the boundary must lie strictly above 4 and less than or equal to 5.

6

Solve the compound inequality for

k

.

23.5 < k \le 29

Performing algebraic operations on all parts of the inequality to isolate

k

.

7

Identify and count all integer solutions for

k

.

6 integers (24, 25, 26, 27, 28, 29)

Counting the integers in the range

(23.5, 29]

yields the final answer.

Key Concept

Solving linear inequalities in one variable with parameter constraints and identifying integer solution sets.

Estimated Time:3m 0s

Question 252Question →

A custom t-shirt printing company charges a one-time design setup fee plus a fixed price for each t-shirt printed. The total cost, $C$ , in dollars, for an order of $n$ t-shirts is given by the equation $C = 15n + 50$ . What does the number $50$ represent in this equation?

The one-time design setup fee, in dollars

The price, in dollars, for each printed t-shirt

The total cost, in dollars, for an order of 1 t-shirt

The number of t-shirts that are printed in an order that has no setup fee

Show answer & explanation

Answer: The one-time design setup fee, in dollars

Answer

The one-time design setup fee, in dollars

The constant term

50

in the linear equation

C = 15n + 50

represents the initial value of the cost function when no t-shirts are ordered (

n = 0

). In this context, this represents the one-time design setup fee.

Step-by-Step Solution

1

Identify the structure of the linear equation

C = 15n + 50

.

The equation is in the slope-intercept form

y = mx + b

, where

C

is the total cost,

n

is the number of t-shirts,

15

is the slope, and

50

is the y-intercept.

Analyzing the linear equation structure helps map its parts to the contextual scenario.

2

Interpret the constant term

50

within the context.

The constant term represents the value of

C

when

n = 0

, which is

C = 15(0) + 50 = 50

dollars.

When no t-shirts are printed, the only cost incurred is the initial flat setup fee.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 253Question →

In the $xy$ -plane, a point with coordinates $(a, b)$ lies in the solution set of the system of inequalities below:

\begin{aligned} y &\ge -2x + 8 \\ y &< \frac{1}{2}x + 3 \end{aligned}

If $a$ and $b$ are integers, what is the minimum possible value of $a$ ?

1

2

3

4

Show answer & explanation

Answer: 3

Answer

3

The correct value is 3. Substituting

x = 3

into the system gives

y \ge 2

and

y < 4.5

. Since the coordinate

b

must be an integer,

b

can be 2, 3, or 4, showing that a valid integer coordinate pair exists. For any integer

x < 3

, there are no integer solutions for

y

.

Step-by-Step Solution

1

Find the intersection point of the boundary lines of the system of inequalities by setting their corresponding equations equal.

-2x + 8 = \frac{1}{2}x + 3 \implies 5 = 2.5x \implies x = 2

This determines the x-coordinate where the boundaries of the two inequality regions intersect.

2

Evaluate the system of inequalities at the intersection point

x = 2

.

y \ge -2(2) + 8 \implies y \ge 4

and

y < \frac{1}{2}(2) + 3 \implies y < 4

. This yields

4 \le y < 4

, which has no solution.

This checks if any valid solution exists at the boundary intersection point.

3

Test consecutive integer values of

x

greater than 2 to find the first value that allows at least one integer

y

.

Substituting

x = 3

gives

y \ge -2(3) + 8 \implies y \ge 2

and

y < \frac{1}{2}(3) + 3 \implies y < 4.5

. The integers that satisfy

2 \le y < 4.5

are

y = 2, 3,

and

4

.

Since

a = 3

is the smallest integer greater than 2 that yields integer solutions for

y

, it is the minimum possible value.

Key Concept

Identifying integer coordinate solutions in systems of linear inequalities

Estimated Time:2m 0s

Question 254Question →

An oceanographic research vessel measures hydrostatic pressure during deep-sea dives. During a dive in temperate waters, the pressure $P$ , in atmospheres (atm), at a depth of $d$ meters below the surface is modeled by the linear equation $P = \rho d + 1.0$ , where $\rho$ is a constant representing the rate of change of pressure with respect to depth. At a depth of $150$ meters, the pressure is $16.0$ atm. During a second dive in a high-salinity region, the rate of change of pressure with respect to depth is $25\%$ greater than the rate in the temperate waters. According to the model for the second dive, at what depth, in meters, is the pressure $26.0$ atm?

Show answer & explanation

Answer: 200

Answer

200

To find the depth on the second dive where the pressure is

26.0

atm, we first find the slope of the linear model for the first dive. Given

P = \rho d + 1.0

, substituting the point

(150, 16.0)

gives

16.0 = 150\rho + 1.0

, which yields

\rho = 0.1

atm/m. The slope for the second dive is

25\%

greater, so the new slope is

0.1 \times 1.25 = 0.125

atm/m. The equation for the second dive is

P = 0.125d + 1.0

. Substituting

P = 26.0

yields

26.0 = 0.125d + 1.0

. Solving for

d

gives

25.0 = 0.125d

, which results in a depth of

200

meters.

Step-by-Step Solution

1

Find the rate of change of pressure with respect to depth in temperate waters.

\rho = 0.1

atm/m

Substitute the depth

d = 150

and pressure

P = 16.0

into the linear equation

P = \rho d + 1.0

to get

16.0 = 150\rho + 1.0

. Solving for

\rho

yields

15.0 = 150\rho

, which means

\rho = 0.1

.

2

Calculate the rate of change of pressure with respect to depth for the second dive.

\rho_{\text{new}} = 0.125

atm/m

The rate of change for the second dive is

25\%

greater than the rate of

0.1

atm/m from the first dive. Thus,

\rho_{\text{new}} = 0.1 \times 1.25 = 0.125

.

3

Calculate the depth where the pressure is

26.0

atm using the new rate of change.

d = 200

meters

Set up the equation for the second dive:

P = 0.125d + 1.0

. Substitute

P = 26.0

to get

26.0 = 0.125d + 1.0

. Subtracting

1.0

from both sides gives

25.0 = 0.125d

. Solving for

d

gives

d = \frac{25.0}{0.125} = 200

.

Key Concept

Interpreting and manipulating slope in a linear relationship context

Question 255Question →

A scientist is monitoring the volume of liquid nitrogen in a storage tank. The volume $V$ , in liters, of liquid nitrogen remaining in the tank $t$ hours after a cooling system malfunction is modeled by the equation $V = 240 - 1.5t$ . After how many hours of malfunction will there be exactly $180$ liters of liquid nitrogen remaining in the tank?

Show answer & explanation

Answer: 40

Answer

The cooling system malfunction must last 40 hours for there to be exactly 180 liters of liquid nitrogen remaining in the tank.

To find the number of hours of malfunction until

180

liters of liquid nitrogen remain in the tank, substitute

180

for

V

in the given model equation, yielding

180 = 240 - 1.5t

. Subtracting

240

from both sides of the equation results in

-60 = -1.5t

. Dividing both sides by

-1.5

gives the final value of

t = 40

hours.

Step-by-Step Solution

1

Substitute the target volume of liquid nitrogen into the linear equation.

180 = 240 - 1.5t

The question asks for the time

t

when the remaining volume

V

is exactly

180

liters.

2

Isolate the term containing the variable by subtracting the initial constant volume from both sides.

-60 = -1.5t

Subtracting

240

from both sides of the equation begins the process of isolating the variable

t

.

3

Divide both sides of the equation by the rate coefficient to solve for time.

t = 40

Dividing by

-1.5

isolates the variable

t

and gives the solution in hours.

Key Concept

Interpreting values and solving equations in linear contexts

Estimated Time:45s

Question 256Question →

A line has the equation $y = mx + k$ in the $xy$ -plane, where $m$ and $k$ are constants. If the line passes through the points $(2, p)$ and $(p, -4)$ , and has a $y$ -intercept of $(0, 3p)$ for some constant $p > 0$ , what is the slope, $m$ , of the line?

-4

4

-1

1

Show answer & explanation

Answer: $-4$

Answer

-4

The correct answer is

-4

. Substituting the

y

-intercept

(0, 3p)

into the slope-intercept form equation

y = mx + k

gives

k = 3p

. Substituting the point

(2, p)

into

y = mx + 3p

gives

p = 2m + 3p

, which simplifies to

m = -p

. Then, substituting the point

(p, -4)

into

y = mx + 3p

gives

-4 = mp + 3p

. Replacing

m

with

-p

results in

-4 = -p^2 + 3p

, which can be rearranged to the quadratic equation

p^2 - 3p - 4 = 0

. Factoring this equation yields

(p - 4)(p + 1) = 0

. Since

p > 0

, the only valid solution is

p = 4

. Substituting

p = 4

back into the expression for

m

gives the slope

m = -4

.

Step-by-Step Solution

1

Determine the y-intercept constant

k

in terms of

p

.

k = 3p

The equation of the line is

y = mx + k

. Since the

y

-intercept is given as

(0, 3p)

, substituting

x = 0

and

y = 3p

gives

k = 3p

.

2

Find the relationship between the slope

m

and the parameter

p

.

m = -p

Since the line passes through

(2, p)

, we substitute these coordinates into the equation

y = mx + 3p

to get

p = 2m + 3p

. Solving for

m

yields

2m = -2p

, or

m = -p

.

3

Substitute the point

(p, -4)

into the linear equation and solve the resulting quadratic equation for

p

.

p = 4

Substituting the coordinates of

(p, -4)

into

y = mx + 3p

gives

-4 = mp + 3p

. Substituting

m = -p

gives

-4 = -p^2 + 3p

, which simplifies to the quadratic equation

p^2 - 3p - 4 = 0

. Factoring this equation gives

(p - 4)(p + 1) = 0

. Since

p > 0

, we must have

p = 4

.

4

Calculate the slope

m

.

m = -4

Since

m = -p

and we found

p = 4

, the slope is

m = -4

.

Key Concept

Determining the slope of a line from given points and a parameter in a linear equation.

Alternative Method

Instead of finding the equation of the line first, the slope

m

can be expressed directly using the slope formula between the points

(2, p)

and

(p, -4)

:

m = \frac{-4 - p}{p - 2}

. Since the slope of the line connecting

(0, 3p)

and

(2, p)

is also

m = \frac{p - 3p}{2 - 0} = -p

, we can set these two expressions for

m

equal to each other:

-p = \frac{-4 - p}{p - 2}

. Multiplying both sides by

p - 2

yields

-p(p - 2) = -4 - p \implies -p^2 + 2p = -4 - p

. Rearranging the terms gives the quadratic equation

p^2 - 3p - 4 = 0

. Factoring gives

(p - 4)(p + 1) = 0

. Since

p > 0

, we find

p = 4

, and the slope is

m = -p = -4

.

Estimated Time:2m 0s

Question 257Question →

A manufacturing company uses the equation below to model the total daily cost, $K$ , in thousands of dollars, of operating a factory when $x$ units are produced:

K = 0.08(x - 150) + 22

where $150 \le x \le 800$ . The company plans to transition to a new production setup that will increase the daily fixed operating cost by $\$3,000$ but will decrease the cost to produce each unit by $\$20$ . Which of the following equations best models the new total daily cost, $K_{\text{new}}$ , in thousands of dollars, to produce $x$ units under the new setup?

K_{\text{new}} = 0.06x + 13

K_{\text{new}} = 0.06x + 25

K_{\text{new}} = 0.08x + 25

K_{\text{new}} = 0.06x + 10

Show answer & explanation

Answer: $K_{\text{new}} = 0.06x + 13$

Answer

The equation that best models the new total daily cost is

K_{\text{new}} = 0.06x + 13

.

The correct equation is determined by first expressing the original cost equation in slope-intercept form:

K = 0.08x + 10

. This reveals that the original variable cost is

0.08

thousand dollars (

\$80

) per unit and the original fixed cost is

10

thousand dollars (

\$10,000

). Decreasing the variable cost by

\$20

(

0.02

thousand dollars) results in a new slope of

0.06

. Increasing the fixed cost by

\$3,000

(

3

thousand dollars) yields a new y-intercept of

13

. Thus, the new cost model is

K_{\text{new}} = 0.06x + 13

.

Step-by-Step Solution

1

Expand the original equation to identify the baseline fixed and variable costs.

K = 0.08x - 12 + 22 \implies K = 0.08x + 10

Converting the equation to slope-intercept form,

y = mx + b

, allows direct identification of the variable cost per unit (slope,

m

) and the fixed cost (y-intercept,

b

).

2

Convert the baseline parameters to their corresponding contextual values in dollars.

Variable cost:

0.08

thousand dollars =

\$80

per unit. Fixed cost:

10

thousand dollars =

\$10,000

daily.

Since the cost

K

is in thousands of dollars, a coefficient of

0.08

represents

0.08 \times 1,000 = \$80

per unit, and a constant of

10

represents

10 \times 1,000 = \$10,000

.

3

Apply the described operational changes to determine the new variable and fixed costs.

New variable cost:

\$80 - \$20 = \$60

per unit, or

0.06

thousand dollars. New fixed cost:

\$10,000 + \$3,000 = \$13,000

daily, or

13

thousand dollars.

The unit cost decreases by

\$20

, lowering it from

\$80

to

\$60

. The fixed cost increases by

\$3,000

, raising it from

\$10,000

to

\$13,000

.

4

Assemble the new linear equation using the updated parameters.

K_{\text{new}} = 0.06x + 13

Combining the new variable cost per unit (

0.06

) and the new fixed cost (

13

) yields the new daily cost equation.

Key Concept

Converting and modifying linear equations in a real-world context by identifying and altering slopes and y-intercepts.

Estimated Time:3m 0s

Question 258Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has the property that the value of $y$ decreases by $3$ for every increase of $5$ in the value of $x$ . If the graph of this equation passes through the point $(4, -2)$ and $a = 6$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is

4

.

The correct answer is

4

. The slope of the line described is

m = \frac{\Delta y}{\Delta x} = \frac{-3}{5} = -0.6

. Rearranging

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows that the slope is

-\frac{a}{b}

. Setting these equal gives

-\frac{a}{b} = -\frac{3}{5} \implies \frac{a}{b} = \frac{3}{5}

. Since

a = 6

, we find

b = 10

. Substituting

a = 6

,

b = 10

, and the coordinates of the point

(4, -2)

into the original equation yields

6(4) + 10(-2) = 4

.

Step-by-Step Solution

1

Determine the slope of the line from the described relationship between

x

and

y

.

The slope

m

of the line is

-\frac{3}{5}

.

The slope represents the change in

y

divided by the change in

x

. Since

y

decreases by

3

(

\Delta y = -3

) for every increase of

5

in

x

(

\Delta x = 5

), the slope is

m = \frac{\Delta y}{\Delta x} = -\frac{3}{5}

.

2

Express the slope of the line in terms of the coefficients from the standard form equation

ax + by = c

.

The slope of the line is

-\frac{a}{b}

.

Rewriting the equation

ax + by = c

in slope-intercept form gives

by = -ax + c \implies y = -\frac{a}{b}x + \frac{c}{b}

. The coefficient of

x

is the slope, so

m = -\frac{a}{b}

.

3

Equate the two expressions for the slope and solve for

b

using the given value of

a = 6

.

b = 10

Setting

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

gives

\frac{a}{b} = \frac{3}{5}

. Substituting

a = 6

yields

\frac{6}{b} = \frac{3}{5}

, which simplifies to

3b = 30

, so

b = 10

.

4

Substitute the point

(4, -2)

and the values of

a

and

b

into the equation

ax + by = c

to solve for

c

.

c = 4

Substituting

a = 6

,

b = 10

,

x = 4

, and

y = -2

into

ax + by = c

gives

6(4) + 10(-2) = c \implies 24 - 20 = c \implies c = 4

.

Key Concept

Linear Equations in Two Variables

Question 259Question →

A bakery sells blueberry muffins and bran muffins. The total cost of $6$ blueberry muffins and $4$ bran muffins is $\$22.00$ . The total cost of $5$ blueberry muffins and $8$ bran muffins is $\$26.50$ . What is the cost, in dollars, of $1$ blueberry muffin?

Show answer & explanation

Answer: 2.5

Answer

The cost of one blueberry muffin is 2.5 dollars.

The correct answer is 2.5. By setting up the system of linear equations

6x + 4y = 22

and

5x + 8y = 26.5

, where

x

represents the cost of one blueberry muffin and

y

represents the cost of one bran muffin, we can solve for

x

. Multiplying the first equation by 2 gives

12x + 8y = 44

. Subtracting the second equation,

5x + 8y = 26.5

, from the modified first equation yields

7x = 17.5

. Dividing by 7 results in

x = 2.5

.

Step-by-Step Solution

1

Set up a system of equations.

6x + 4y = 22

and

5x + 8y = 26.5

Translate the context into mathematical statements where

x

is the price of a blueberry muffin and

y

is the price of a bran muffin.

2

Multiply the first equation by 2.

12x + 8y = 44

Align the coefficients of

y

to facilitate elimination.

3

Subtract the second equation from the modified first equation.

7x = 17.5

Eliminate the variable

y

to solve for

x

.

4

Divide by 7.

x = 2.5

Isolate the variable

x

to find the cost of a blueberry muffin.

Key Concept

Solving systems of two linear equations using elimination.

Alternative Method

Alternatively, solve for

y

in terms of

x

using the first equation:

y = 5.5 - 1.5x

. Substitute this expression into the second equation:

5x + 8(5.5 - 1.5x) = 26.5

, which simplifies to

5x + 44 - 12x = 26.5 \\Rightarrow -7x = -17.5 \\Rightarrow x = 2.5

.

Estimated Time:1m 30s

Question 260Question →

If $\frac{3}{5}(5x - 10) - \frac{1}{2}(4x - 6) = 12$ , what is the value of $2x - 5$ ?

13

15

25

37

Show answer & explanation

Answer: 25

Answer

The correct answer is 25.

The correct answer is 25. Distributing the fractions yields

3x - 6 - 2x + 3 = 12

, which simplifies to

x - 3 = 12

, so

x = 15

. Substituting this into

2x - 5

gives

2(15) - 5 = 25

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses.

(3x - 6) - (2x - 3) = 12

To simplify the equation by removing the parentheses.

2

Combine like terms on the left side of the equation.

x - 3 = 12

To group the variable terms and constant terms together.

3

Isolate the variable

x

by adding 3 to both sides of the equation.

x = 15

To find the value of

x

.

4

Substitute

x = 15

into the expression

2x - 5

.

2(15) - 5 = 25

To find the final value requested by the question.

Key Concept

Solving linear equations in one variable with grouping symbols and fractions, and evaluating an algebraic expression.

Estimated Time:1m 30s

Algebra

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