Algebra Practice Questions — SAT — Page 12

Question 221Question →

Oven A preheats at a constant rate of $15$ degrees Celsius per minute, starting from an initial temperature of $25$ degrees Celsius. Oven B starts preheating $5$ minutes after Oven A begins, starting from an initial temperature of $20$ degrees Celsius and preheating at a constant rate of $20$ degrees Celsius per minute. If both ovens continue to preheat, how many minutes after Oven A begins preheating will both ovens reach the same temperature?

Show answer & explanation

Answer: 21

Answer

The ovens will reach the same temperature

21

minutes after Oven A begins preheating.

To find the number of minutes after Oven A begins preheating when both ovens reach the same temperature, we can write an equation in terms of

t

, the time in minutes since Oven A started. Oven A starts at

25

degrees Celsius and increases by

15

degrees per minute, so its temperature is

25 + 15t

. Oven B starts

5

minutes later, meaning it preheats for

t - 5

minutes. Starting from

20

degrees Celsius and preheating at

20

degrees per minute, Oven B's temperature is

20 + 20(t - 5)

. Setting these two expressions equal gives the equation

25 + 15t = 20 + 20(t - 5)

. Distributing

20

yields

25 + 15t = 20t - 80

. Isolating

t

gives

5t = 105

, which results in

t = 21

.

Step-by-Step Solution

1

Set up expressions representing the temperature of each oven

t

minutes after Oven A begins preheating.

Oven A:

25 + 15t

; Oven B:

20 + 20(t - 5)

Since Oven B starts

5

minutes after Oven A, it has been preheating for

t - 5

minutes.

2

Set the temperature expressions equal to find the time at which they reach the same temperature.

25 + 15t = 20 + 20(t - 5)

We want to find the value of

t

where the temperatures of the two ovens are equal.

3

Distribute and simplify the equation.

25 + 15t = 20t - 80

Distributing the

20

across

(t - 5)

yields

20t - 100

. Combining the constant terms gives

20 - 100 = -80

.

4

Solve for

t

by isolating the variable term.

5t = 105

, which gives

t = 21

Subtract

15t

and add

80

to both sides, then divide by

5

.

Key Concept

Solving linear equations in one variable that model real-world situations with a time delay.

Question 222Question →

Consider the system of linear equations below, where $k$ is a constant:

\begin{aligned} ky - 3x &= 4 \\ 4y - (k-1)x &= 3 \end{aligned}

If the system of equations has no solution, what is the sum of all possible values of $k$ ?

1

-1

4

7

Show answer & explanation

Answer: 1

Answer

1

For the system of equations to have no solution, the lines must be parallel. In slope-intercept form, the equations are

y = \frac{3}{k}x + \frac{4}{k}

and

y = \frac{k-1}{4}x + \frac{3}{4}

. Setting the slopes equal gives

\frac{3}{k} = \frac{k-1}{4}

, which simplifies to the quadratic equation

k^2 - k - 12 = 0

. Factoring this equation yields

(k-4)(k+3) = 0

, giving

k = 4

and

k = -3

. Since both values yield different y-intercepts for the two lines, they both result in parallel lines with no intersection. The sum of these values is

4 + (-3) = 1

.

Step-by-Step Solution

1

Express both equations in slope-intercept form (

y = mx + b

) to find their slopes.

The first equation becomes

y = \frac{3}{k}x + \frac{4}{k}

(for

k \neq 0

). The second equation becomes

y = \frac{k-1}{4}x + \frac{3}{4}

.

For a system of linear equations to have no solution, the lines must be parallel, meaning they have the same slope but different y-intercepts.

2

Set the slopes of the two lines equal to each other to solve for

k

.

\frac{3}{k} = \frac{k-1}{4} \implies k(k-1) = 12 \implies k^2 - k - 12 = 0

.

Equating the slopes allows us to find the values of

k

that make the lines parallel.

3

Solve the quadratic equation

k^2 - k - 12 = 0

by factoring.

(k-4)(k+3) = 0 \implies k = 4

or

k = -3

.

This determines the specific values of

k

that make the slopes equal.

4

Verify that both values of

k

produce different y-intercepts (so the lines do not coincide) and calculate their sum.

For

k = 4

, the y-intercepts are

\frac{4}{4} = 1

and

\frac{3}{4}

, which are different. For

k = -3

, the y-intercepts are

-\frac{4}{3}

and

\frac{3}{4}

, which are also different. The sum of the values of

k

is

4 + (-3) = 1

.

We must confirm the lines are parallel and not identical, then compute the required sum.

Key Concept

Systems of linear equations with no solution represent parallel lines with equal slopes and unequal y-intercepts.

Question 223Question →

A community garden has a total area of $1,200$ square feet available for planting tomatoes and squash. Each tomato plant requires $8$ square feet of space, and each squash plant requires $15$ square feet of space. The garden coordinators want to plant at least $100$ plants in total, but they want the number of squash plants to be no more than twice the number of tomato plants. If $t$ represents the number of tomato plants and $s$ represents the number of squash plants, which of the following systems of inequalities represents this situation?

\begin{cases} 8t + 15s \geq 1,200 \\ t + s \leq 100 \\ s \leq 2t \end{cases}

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ t \leq 2s \end{cases}

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \geq 2t \end{cases}

Show answer & explanation

Answer: $\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}$

Answer

The system of inequalities containing the constraints

8t + 15s \leq 1,200

,

t + s \geq 100

, and

s \leq 2t

.

The correct system of inequalities maps the garden's constraints accurately:

8t + 15s \leq 1,200

limits the space usage to a maximum of

1,200

square feet;

t + s \geq 100

ensures there are at least

100

plants in total; and

s \leq 2t

restricts the number of squash plants to at most twice the number of tomato plants.

Step-by-Step Solution

1

Translate the area constraint.

8t + 15s \leq 1,200

Since the total area available is

1,200

square feet and each tomato plant requires

8

square feet while each squash plant requires

15

square feet, the combined space used by

t

tomato plants and

s

squash plants cannot exceed

1,200

.

2

Translate the total plant count constraint.

t + s \geq 100

The coordinators want at least

100

plants, meaning the sum of the tomato and squash plants must be greater than or equal to

100

.

3

Translate the plant ratio constraint.

s \leq 2t

The number of squash plants (

s

) must be no more than (less than or equal to) twice the number of tomato plants (

2t

).

4

Combine the inequalities into a single system.

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}

Grouping these three constraints forms the mathematical model that represents the given scenario.

Key Concept

Translating real-world constraints into a system of linear inequalities in two variables.

Question 224Question →

For each real number

k

except

2

, the system of equations below has a unique solution

(x, y)

.

\begin{aligned} kx + (k + 2)y &= 3k + 1 \\ (k - 1)x + ky &= 2k - 1 \end{aligned}

If the solution

(x, y)

to the system also satisfies the equation

x + 2y = 6

, what is the value of

k

?

Show answer & explanation

Answer: 4

Answer

The value of

k

is 4.

Subtracting the second equation from the first equation yields

x + 2y = k + 2

. Since the solution must also satisfy

x + 2y = 6

, we set

k + 2 = 6

, which gives

k = 4

.

Step-by-Step Solution

1

Subtract the second equation from the first equation.

x + 2y = k + 2

To find a direct linear combination of

x

and

y

that can be compared directly to the target equation.

2

Equate the resulting expression to the target equation

x + 2y = 6

.

k + 2 = 6

Since the solution

(x, y)

must satisfy

x + 2y = 6

, the value of the linear combination

x + 2y

from the system must equal

6

.

3

Solve for

k

by subtracting 2 from both sides.

k = 4

Isolating the variable

k

yields the final solution.

Key Concept

Solving systems of linear equations with parameter coefficients by identifying algebraic structure and linear combinations.

Alternative Method

Solve the system for

x

and

y

in terms of

k

using elimination. Multiplying the first equation by

(k-1)

and the second by

k

, and then subtracting them yields

y = \frac{k^2 - k - 1}{k - 2}

. Substituting this back gives

x = \frac{-k^2 + 2k - 2}{k - 2}

. Substituting these expressions into the equation

x + 2y = 6

results in

\frac{k^2 - 4}{k - 2} = 6

. For

k \neq 2

, factoring

k^2 - 4

as

(k-2)(k+2)

allows simplification to

k + 2 = 6

, which yields

k = 4

.

Estimated Time:2m 0s

Question 225Question →

A solution to a system of inequalities in the $xy$ -plane is represented by the point $(2, y)$ . If the system is defined by:

y \geq 4x - 2

y \leq -2x + 10

what is the value of $y$ ?

Show answer & explanation

Answer: 6

Answer

The value of

y

must be exactly 6 because it is bounded both below and above by 6.

Substituting

x = 2

into the system of inequalities yields two constraints:

y \geq 4(2) - 2

, which simplifies to

y \geq 6

, and

y \leq -2(2) + 10

, which simplifies to

y \leq 6

. The only value of

y

that satisfies both statements is 6.

Step-by-Step Solution

1

Substitute

x = 2

into the first inequality

y \geq 4x - 2

y \geq 6

To determine the constraint on the

y

-value from the first inequality at the coordinate

x = 2

.

2

Substitute

x = 2

into the second inequality

y \leq -2x + 10

y \leq 6

To determine the constraint on the

y

-value from the second inequality at the coordinate

x = 2

.

3

Find the value of

y

that satisfies both

y \geq 6

and

y \leq 6

y = 6

For a value to be simultaneously greater than or equal to 6 and less than or equal to 6, it must be equal to 6.

Key Concept

Evaluating a system of linear inequalities at a specific coordinate to determine a unique boundary value

Estimated Time:45s

Question 226Question →

In the equation below, $a$ and $b$ are constants.

a(2x - 9) - 3(x - b) = 7x + 12

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

-19

10

11

19

Show answer & explanation

Answer: 19

Answer

19

To find the value of

b

for which the equation has infinitely many solutions, we expand the left side to get

2ax - 9a - 3x + 3b = 7x + 12

. Grouping the terms by variable yields

(2a - 3)x + (3b - 9a) = 7x + 12

. For the equation to have infinitely many solutions, the coefficient of

x

and the constant term on both sides must be equal. Setting the

x

-coefficients equal gives

2a - 3 = 7

, which simplifies to

2a = 10

and

a = 5

. Setting the constant terms equal gives

3b - 9a = 12

. Substituting

a = 5

into this equation yields

3b - 9(5) = 12

, which simplifies to

3b - 45 = 12

. Adding

45

to both sides gives

3b = 57

, which results in

b = 19

.

Step-by-Step Solution

1

Expand and simplify the left side of the equation.

(2a - 3)x + (3b - 9a) = 7x + 12

To compare coefficients with the right side of the equation, we need to group the terms on the left side.

2

Set the coefficients of

x

on both sides equal to each other to solve for

a

.

2a - 3 = 7 \implies 2a = 10 \implies a = 5

For an equation to have infinitely many solutions, the coefficients of the variable on both sides must be equal.

3

Set the constant terms on both sides equal to each other, substitute the value of

a

, and solve for

b

.

3b - 9a = 12 \implies 3b - 9(5) = 12 \implies 3b - 45 = 12 \implies 3b = 57 \implies b = 19

For an equation to have infinitely many solutions, the constant terms on both sides must also be equal.

Key Concept

Linear equations with infinitely many solutions require the coefficients of the variable to be equal and the constant terms to be equal on both sides of the equation.

Question 227Question →

A delivery truck carries a total of $x$ bags of organic fertilizer and $y$ bags of compost. Each bag of fertilizer weighs $45$ pounds, and each bag of compost weighs $30$ pounds. When the truck is loaded to exactly $80\%$ of its maximum payload capacity, the relationship between $x$ and $y$ can be represented by the equation $3x + 2y = 240$ . What is the truck's maximum payload capacity, in pounds?

2,880

3,000

3,600

4,500

Show answer & explanation

Answer: $4,500$

Answer

The maximum payload capacity of the truck is

4,500

pounds.

The total weight of the bags loaded on the truck is represented by the expression

45x + 30y

pounds. Factoring out

15

from this expression gives

15(3x + 2y)

. We are given that when the truck is loaded to exactly

80\%

of its capacity, the relationship between

x

and

y

is

3x + 2y = 240

. Substituting

240

for

3x + 2y

in the weight expression shows that the weight of the load at

80\%

capacity is

15 \times 240 = 3,600

pounds. To find the maximum capacity, we set up the equation

0.80 \times C = 3,600

, where

C

is the maximum payload capacity. Solving for

C

yields

C = \frac{3,600}{0.80} = 4,500

pounds.

Step-by-Step Solution

1

Write the expression for the total weight of the bags in terms of

x

and

y

.

Total weight

W = 45x + 30y

pounds.

Each of the

x

bags of fertilizer weighs

45

pounds and each of the

y

bags of compost weighs

30

pounds.

2

Relate the total weight expression to the given equation for

80\%

capacity.

45x + 30y = 15(3x + 2y)

Factoring out the greatest common divisor of

45

and

30

, which is

15

, allows us to write the weight expression in terms of the left-hand side of the given equation

3x + 2y = 240

.

3

Calculate the actual weight of the load when the truck is at

80\%

capacity.

W = 15 \times 240 = 3,600

pounds.

Since the relationship at

80\%

capacity is

3x + 2y = 240

, we can substitute

240

for

3x + 2y

in the factored weight expression.

4

Calculate the maximum payload capacity of the truck.

Maximum capacity

C = \frac{3,600}{0.80} = 4,500

pounds.

The weight of

3,600

pounds represents

80\%

of the maximum payload capacity, so dividing the actual weight by

0.80

gives the total capacity.

Key Concept

Translating a real-world scenario into a linear equation and interpreting its coefficients and constants to solve for unknown quantities.

Question 228Question →

In the

xy

-plane, a point with coordinates

(x, y)

lies in the solution set of the system of inequalities below:

\begin{aligned} 2(x - 5) + y &\leq 16 \\ x + y &\leq 25 \end{aligned}

If

y \geq 8

, what is the maximum possible integer value of

x

?

6

9

13

17

Show answer & explanation

Answer: 9

Answer

9

To find the maximum possible integer value of

x

in the system's solution set under the constraint

y \geq 8

, we find the boundaries for

x

in both inequalities. Simplifying the first inequality,

2(x - 5) + y \leq 16

, gives

2x - 10 + y \leq 16

, which simplifies to

x \leq 13 - 0.5y

. The second inequality,

x + y \leq 25

, simplifies to

x \leq 25 - y

. Since both boundary lines have negative slopes, the maximum value of

x

for any point in the solution set occurs at the minimum value of

y

, which is

y = 8

. Substituting

y = 8

into both simplified inequalities gives

x \leq 9

and

x \leq 17

. To satisfy both inequalities simultaneously,

x

must be less than or equal to the more restrictive upper bound, which is 9. Thus, the maximum possible integer value of

x

is 9.

Step-by-Step Solution

1

Simplify the first inequality to isolate x.

2(x - 5) + y \leq 16 \implies 2x - 10 + y \leq 16 \implies 2x \leq 26 - y \implies x \leq 13 - 0.5y

To determine the boundary condition imposed on x by the first inequality.

2

Simplify the second inequality to isolate x.

x + y \leq 25 \implies x \leq 25 - y

To determine the boundary condition imposed on x by the second inequality.

3

Determine the maximum value of x under the condition

y \geq 8

.

For the first inequality, the maximum value of x occurs at the minimum value of y (

y = 8

), which gives

x \leq 13 - 0.5(8) = 9

. For the second inequality, the maximum value of x occurs at

y = 8

, which gives

x \leq 25 - 8 = 17

.

Because both boundary lines have negative slopes, the maximum value of x is achieved when y is at its minimum.

4

Find the intersection of the constraints to find the maximum possible value of x that satisfies both inequalities.

Since x must be less than or equal to 9 and less than or equal to 17, the maximum integer value that satisfies both conditions is 9.

The solution set must satisfy all inequalities in the system simultaneously.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 229Question →

A tutor charges a total fee, $C$ , in dollars, for a tutoring session of $h$ hours according to the equation $C = 25h + 40$ . What is the best interpretation of $40$ in this context?

The fee, in dollars, charged for each hour of tutoring

The total fee, in dollars, charged for a 1-hour tutoring session

The flat sign-up fee, in dollars, charged by the tutor

The number of hours of tutoring for which the total fee is $0

Show answer & explanation

Answer: The flat sign-up fee, in dollars, charged by the tutor

Answer

The flat sign-up fee, in dollars, charged by the tutor

The constant term in the linear equation represents the value of the dependent variable when the independent variable is zero. In this context, when the tutoring hours are zero, the total fee is 40 dollars, representing the flat sign-up fee.

Step-by-Step Solution

1

Identify the structure of the linear equation

The equation

C = 25h + 40

matches the slope-intercept form

y = mx + b

.

The linear model consists of a variable term representing the hourly rate and a constant term representing the flat fee.

2

Determine the value of the constant term when the variable is zero

When

h = 0

, the equation simplifies to

C = 40

.

Setting the hours of tutoring to zero isolates the starting value of the function.

3

Interpret the starting value in the context of the problem

A session of

0

hours corresponds to the cost before any tutoring begins, which represents a flat initial sign-up fee of

40

.

Connecting the mathematical y-intercept to the real-world scenario identifies the interpretation of the constant.

Key Concept

Interpreting Linear Relationships in Context

Question 230Question →

A student works two part-time jobs: tutoring, which pays $20$ dollars per hour, and working at a bookstore, which pays $12$ dollars per hour. The student can work at most $15$ hours per week and wants to earn at least $220$ dollars per week. If the student works a whole number of hours at each job, what is the minimum number of hours the student must tutor to meet these requirements?

Show answer & explanation

Answer: 5

Answer

5

The system of inequalities modeling the scenario consists of

x + y \leq 15

(representing the limit on total hours) and

20x + 12y \geq 220

(representing the earning target), where

x

is the hours of tutoring and

y

is the hours at the bookstore. To find the minimum hours of tutoring, we assume the maximum bookstore hours

y = 15 - x

and substitute it into the earnings equation:

20x + 12(15 - x) \geq 220

. Simplifying yields

8x + 180 \geq 220

, which reduces to

8x \geq 40

, or

x \geq 5

. The minimum integer value that satisfies this condition is 5.

Step-by-Step Solution

1

Define variables and set up the constraint for total hours worked.

x + y \leq 15

Let

x

be the number of hours tutoring and

y

be the number of hours working at the bookstore. The total hours cannot exceed 15.

2

Set up the constraint for the minimum weekly earnings.

20x + 12y \geq 220

Tutoring pays

20

dollars per hour and the bookstore pays

12

dollars per hour, and the total earnings must be at least

220

dollars.

3

Substitute the maximum value of

y

in terms of

x

into the earnings inequality.

20x + 12(15 - x) \geq 220

To minimize

x

, we must maximize

y

. From

x + y \leq 15

, the maximum value of

y

is

15 - x

.

4

Solve the inequality for

x

.

x \geq 5

Distribute and simplify:

20x + 180 - 12x \geq 220 \Rightarrow 8x + 180 \geq 220 \Rightarrow 8x \geq 40 \Rightarrow x \geq 5

.

Key Concept

Solving systems of linear inequalities to find optimal boundary values in context.

Estimated Time:1m 30s

Question 231Question →

Let $c$ be a constant such that $c < 0$ . If $c(x - 3) \leq 4(x + c)$ , which of the following inequalities must be true?

x \leq \frac{7c}{c - 4}

x \geq \frac{c}{c - 4}

x \geq \frac{7c}{c - 4}

x \geq \frac{7c}{c + 4}

Show answer & explanation

Answer: $x \geq \frac{7c}{c - 4}$

Answer

x \geq \frac{7c}{c - 4}

The correct answer is obtained by first distributing terms on both sides to get

cx - 3c \leq 4x + 4c

. Rearranging the terms to isolate the variable

x

on one side yields

(c - 4)x \leq 7c

. Since

c

is a negative constant (

c < 0

), the coefficient

(c - 4)

must also be negative. Dividing both sides of the inequality by this negative coefficient requires reversing the direction of the inequality sign, which yields the final result.

Step-by-Step Solution

1

Distribute the constants on both sides of the inequality.

cx - 3c \leq 4x + 4c

This simplifies the parentheses so that variable terms can be grouped.

2

Group all terms with

x

on the left side and terms with

c

on the right side.

cx - 4x \leq 7c

Isolating the variable terms on one side makes it possible to factor and solve for

x

.

3

Factor out

x

on the left side of the inequality.

(c - 4)x \leq 7c

This expresses the left side as a product of

x

and a single coefficient.

4

Analyze the sign of the coefficient

(c - 4)

given that

c < 0

.

c - 4 < 0

Since

c

is less than

0

, subtracting

4

from

c

must result in a value less than

-4

, which is strictly negative.

5

Divide both sides of the inequality by

(c - 4)

and flip the inequality symbol.

x \geq \frac{7c}{c - 4}

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Solving linear inequalities by isolating the variable, and correctly reversing the inequality direction when multiplying or dividing by a negative variable parameter.

Question 232Question →

A software company offers a Basic subscription for $\$15$ per month and a Premium subscription for $\$30$ per month. In April, the company had a total of $1,200$ active subscriptions. In May, the number of Basic subscriptions increased by $20\%$ , and the number of Premium subscriptions increased by $5\%$ compared to April. If the company's total monthly revenue increased by $\$2,700$ from April to May, how many Premium subscriptions did the company have in April?

Show answer & explanation

Answer: 600

Answer

600

To find the number of Premium subscriptions in April, we can set up a system of linear equations. Let

B

represent the number of Basic subscriptions in April and

P

represent the number of Premium subscriptions in April. Since the total number of subscriptions is

1,200

, we have

B + P = 1,200

. The revenue increase from April to May is

\$2,700

. The increase in Basic subscriptions is

20\%

, which contributes

15 \times 0.20B = 3B

dollars to the revenue increase. The increase in Premium subscriptions is

5\%

, which contributes

30 \times 0.05P = 1.5P

dollars to the revenue increase. Thus,

3B + 1.5P = 2,700

. Substituting

B = 1,200 - P

into this equation yields

3(1,200 - P) + 1.5P = 2,700

. Solving for

P

gives

3,600 - 1.5P = 2,700

, which simplifies to

1.5P = 900

, or

P = 600

.

Step-by-Step Solution

1

Define variables and write the first equation based on the total number of subscriptions in April.

B + P = 1,200

, where

B

is the number of Basic subscriptions and

P

is the number of Premium subscriptions.

To represent the relationship between the two types of subscriptions in April.

2

Write the second equation representing the change in monthly revenue from April to May.

3B + 1.5P = 2,700

The change in revenue is the sum of the increase in revenue from each subscription type:

15(0.20B) + 30(0.05P) = 2,700

.

3

Solve the system of equations by substituting

B = 1,200 - P

into the revenue equation.

3(1,200 - P) + 1.5P = 2,700 \implies 3,600 - 1.5P = 2,700

Substitution eliminates the variable

B

, allowing us to solve for

P

directly.

4

Isolate the variable

P

to find the number of Premium subscriptions in April.

-1.5P = -900 \implies P = 600

Dividing the revenue difference by the coefficient solves for the value of

P

.

Key Concept

Solving systems of linear equations in real-life contexts involving percentage changes.

Question 233Question →

An environmental scientist is monitoring the water level of a reservoir during a dry season. The water level, $L(d)$ , in meters, can be modeled by a linear function of the number of days, $d$ , since the start of the dry season. On day 12, the water level was 30 meters, and on day 20, the water level was 26 meters. If the water level continues to decrease at this constant rate, on which day will the water level be exactly 18 meters?

Day 18

Day 24

Day 36

Day 40

Show answer & explanation

Answer: Day 36

Answer

Day 36

To find the day when the water level is exactly 18 meters, we first find the constant rate of change (slope) using the given points

(12, 30)

and

(20, 26)

. The slope is

m = \frac{26 - 30}{20 - 12} = -0.5

meters per day. Using the point-slope form with

(12, 30)

, we get

L(d) - 30 = -0.5(d - 12)

, which simplifies to

L(d) = -0.5d + 36

. Substituting 18 for

L(d)

gives

18 = -0.5d + 36

. Subtracting 36 from both sides results in

-18 = -0.5d

, and dividing by

-0.5

yields

d = 36

. Thus, the correct answer is Day 36.

Step-by-Step Solution

1

Calculate the constant rate of change (slope,

m

) using the points

(12, 30)

and

(20, 26)

.

m = \frac{26 - 30}{20 - 12} = \frac{-4}{8} = -0.5

meters per day.

A linear function has a constant rate of change, which is determined by the slope formula:

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

.

2

Write the equation of the linear model in point-slope form and solve for the standard form.

L(d) - 30 = -0.5(d - 12) \Rightarrow L(d) = -0.5d + 36

.

Using the point-slope formula

y - y_1 = m(x - x_1)

with the point

(12, 30)

and slope

-0.5

defines the function for any day

d

.

3

Substitute

L(d) = 18

into the equation and solve for

d

.

18 = -0.5d + 36 \Rightarrow -18 = -0.5d \Rightarrow d = 36

.

Setting the dependent variable to the target water level of 18 meters yields the corresponding day.

Key Concept

Determining and evaluating linear equations from two given coordinate points

Alternative Method

Instead of constructing the entire equation, you can analyze the constant rate of change directly. The water level decreases by

4

meters (from 30 to 26) over

8

days (from day 12 to day 20), which means the rate of decrease is

0.5

meters per day. To drop from 26 meters (on day 20) to 18 meters requires a further decrease of

26 - 18 = 8

meters. At a rate of

0.5

meters per day, dropping 8 meters will take

\frac{8}{0.5} = 16

days. Adding these 16 days to day 20 yields day 36.

Estimated Time:1m 30s

Question 234Question →

A shipping company offers two types of delivery services: standard and express. The total shipping cost, in dollars, for a package sent via standard service is modeled by the function $C(w) = 1.25w + b_s$ , where $w$ is the weight of the package, in pounds, and $b_s$ is a constant representing the flat handling fee. The total shipping cost, in dollars, for a package sent via express service is modeled by the function $E(k) = m_e k + b_e$ , where $k$ is the weight of the package, in kilograms, and $b_e$ is a constant representing the flat handling fee.

The rate of change of the express shipping cost with respect to the package's weight, in dollars per kilogram, is $2.4$ times the rate of change of the standard shipping cost with respect to the package's weight, in dollars per pound. The flat handling fee for the express service is $4.50$ dollars more than the flat handling fee for the standard service. If it costs $58.50$ dollars to ship a package weighing $26.4$ pounds using the express service, what is the cost, in dollars, to ship a package weighing $20$ pounds using the standard service? (Assume $1\text{ kilogram} = 2.2\text{ pounds}$ .)

Show answer & explanation

Answer: 43

Answer

The cost to ship a package weighing 20 pounds using the standard service is 43 dollars.

The correct answer is obtained by first calculating the rate of change of the express shipping cost (

2.4 \times 1.25 = 3.00

dollars per kilogram). Next, the weight of the package is converted to kilograms (

26.4 / 2.2 = 12

kilograms) to match the express cost function's variable. Using the given cost of

58.50

dollars for the express shipment, the express handling fee is determined to be

58.50 - 3(12) = 22.50

dollars. The standard handling fee is then found by subtracting

4.50

dollars from the express fee (

22.50 - 4.50 = 18.00

dollars). Finally, the cost of a

20

-pound standard package is computed as

1.25(20) + 18.00 = 43.00

dollars.

Step-by-Step Solution

1

Determine the rate of change for the express service (

m_e

).

m_e = 3.00

dollars per kilogram

The rate of change of the standard shipping cost is

1.25

dollars per pound. Since the rate of change for the express shipping cost is

2.4

times this rate,

m_e = 2.4 \times 1.25 = 3.00

dollars per kilogram.

2

Convert the weight of the express package from pounds to kilograms.

k = 12

kilograms

The weight of the package is given as

26.4

pounds. Using the conversion

1\text{ kilogram} = 2.2\text{ pounds}

, the weight in kilograms is

\frac{26.4}{2.2} = 12

kilograms.

3

Find the express flat handling fee (

b_e

) using the given cost of the express shipment.

b_e = 22.50

dollars

We are given that the cost of shipping a

12

-kilogram package using the express service is

58.50

dollars. Substituting these values into the express cost function:

3(12) + b_e = 58.50 \Rightarrow 36 + b_e = 58.50 \Rightarrow b_e = 22.50

dollars.

4

Find the standard flat handling fee (

b_s

).

b_s = 18.00

dollars

The express handling fee is

4.50

dollars more than the standard handling fee:

b_e = b_s + 4.50 \Rightarrow 22.50 = b_s + 4.50 \Rightarrow b_s = 18.00

dollars.

5

Calculate the cost to ship a

20

-pound package using the standard service.

43.00

dollars

Using the standard cost function

C(w) = 1.25w + b_s

with

w = 20

and

b_s = 18.00

:

C(20) = 1.25(20) + 18.00 = 25.00 + 18.00 = 43.00

dollars.

Key Concept

Interpreting slope, y-intercept, and rates of change of linear functions in a real-world context with unit conversions.

Question 235Question →

Consider the system of equations below.

\begin{aligned} 3(x + y) - 2(x - y) &= 20 \\ 2(x + y) + 3(x - y) &= 22 \end{aligned}

If

(x, y)

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

x

?

Show answer & explanation

Answer: 5

Answer

The value of

x

is

5

.

Expanding and simplifying the given system yields

x + 5y = 20

and

5x - y = 22

. Multiplying the second equation by 5 results in

25x - 5y = 110

. Adding this to the first equation eliminates

y

, leaving

26x = 130

. Dividing both sides by 26 gives

x = 5

.

Step-by-Step Solution

1

Expand the algebraic expressions in both equations to rewrite the system in standard form.

x + 5y = 20

and

5x - y = 22

Expanding the first equation gives

3x + 3y - 2x + 2y = 20

, which simplifies to

x + 5y = 20

. Expanding the second equation gives

2x + 2y + 3x - 3y = 22

, which simplifies to

5x - y = 22

.

2

Multiply the second equation by 5 to align the coefficients of the y-terms.

25x - 5y = 110

Multiplying

5x - y = 22

by 5 allows the subtraction of

y

to cancel with the addition of

5y

in the first equation.

3

Add the first equation to the modified second equation to eliminate the y-variable and solve for x.

26x = 130

, which simplifies to

x = 5

.

Adding

(x + 5y) + (25x - 5y) = 20 + 110

yields

26x = 130

. Dividing by 26 gives the final value of

x

.

Key Concept

Solving systems of linear equations using expansion and elimination.

Alternative Method

Define substitution variables

u = x + y

and

v = x - y

. The system simplifies to

3u - 2v = 20

and

2u + 3v = 22

. Multiplying the first equation by 3 and the second by 2 gives

9u - 6v = 60

and

4u + 6v = 44

. Adding these equations yields

13u = 104 \implies u = 8

. Substituting

u = 8

back in gives

16 + 3v = 22 \implies 3v = 6 \implies v = 2

. Now, solve the system

x + y = 8

and

x - y = 2

. Adding these two equations gives

2x = 10 \implies x = 5

.

Estimated Time:1m 30s

Question 236Question →

A manufacturer produces standard chairs and deluxe chairs. Each standard chair requires $2$ hours of assembly and $1$ hour of finishing. Each deluxe chair requires $3$ hours of assembly and $2$ hours of finishing. The manufacturer has a maximum of $240$ hours available for assembly and a maximum of $150$ hours available for finishing each day. Additionally, the manufacturer must produce at least $10$ deluxe chairs daily. If $x$ represents the number of standard chairs produced daily and $y$ represents the number of deluxe chairs produced daily, which of the following systems of inequalities best represents this situation?

\begin{aligned} 2x + 3y &\geq 240 \\ x + 2y &\geq 150 \\ y &\geq 10 \end{aligned}

\begin{aligned} 3x + 2y &\leq 240 \\ 2x + y &\leq 150 \\ y &\geq 10 \end{aligned}

\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\geq 10 \end{aligned}

\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\leq 10 \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\geq 10 \end{aligned}$

Answer

The system containing

2x + 3y \leq 240

,

x + 2y \leq 150

, and

y \geq 10

.

The correct system represents each constraint accurately. The assembly time requirement is

2x + 3y \leq 240

because standard chairs consume

2

hours each, deluxe chairs consume

3

hours each, and the total cannot exceed

240

. The finishing time requirement is

x + 2y \leq 150

since standard chairs consume

1

hour each, deluxe chairs consume

2

hours each, and the total cannot exceed

150

. Finally, the requirement to produce at least

10

deluxe chairs means

y \geq 10

.

Step-by-Step Solution

1

Set up the inequality for assembly time.

2x + 3y \leq 240

Each standard chair (

x

) needs

2

hours and each deluxe chair (

y

) needs

3

hours, with a maximum limit of

240

hours.

2

Set up the inequality for finishing time.

x + 2y \leq 150

Each standard chair (

x

) needs

1

hour and each deluxe chair (

y

) needs

2

hours, with a maximum limit of

150

hours.

3

Set up the inequality for the minimum production limit of deluxe chairs.

y \geq 10

The manufacturer must produce at least

10

deluxe chairs, meaning the quantity must be greater than or equal to

10

.

4

Combine the individual inequalities into a single system.

The final system is

2x + 3y \leq 240

,

x + 2y \leq 150

, and

y \geq 10

.

All three conditions must be satisfied simultaneously.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 237Question →

A worker at a distribution center packages boxes at a constant rate. The total number of boxes, $B$ , the worker has packaged $h$ hours after starting their shift can be modeled by the equation $B = 12h + 15$ . According to the model, how many boxes were already packaged at the start of the worker's shift?

Show answer & explanation

Answer: 15

Answer

At the start of the worker's shift,

15

boxes were already packaged.

In the linear model

B = 12h + 15

, the term

15

is the constant term (y-intercept), which represents the value of

B

when

h = 0

. In this context,

h = 0

represents the start of the worker's shift. Therefore,

15

boxes were already packaged at the start of the shift.

Step-by-Step Solution

1

Identify the value of

h

that represents the start of the shift.

h = 0

The variable

h

represents the number of hours since the shift started, so the start of the shift corresponds to

0

hours.

2

Substitute

h = 0

into the given equation to find the value of

B

.

B = 15

Evaluating the equation at

h = 0

gives the initial number of packaged boxes, which is represented by the constant term of the linear equation.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 238Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has a slope of $\frac{2}{3}$ and passes through the point $(6, 5)$ . If $a + b = 5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

15

Rearranging the linear equation

ax + by = c

into slope-intercept form

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

shows the slope is

-\frac{a}{b}

. Setting

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

gives

2b = -3a

, or

b = -1.5a

. Substituting this into

a + b = 5

yields

a - 1.5a = 5

, which simplifies to

-0.5a = 5

, so

a = -10

. This means

b = -1.5(-10) = 15

. Substituting the values of

a = -10

and

b = 15

along with the point

(6, 5)

into the equation

ax + by = c

yields

(-10)(6) + (15)(5) = 15

, so the value of

c

is

15

.

Step-by-Step Solution

1

Rewrite the standard form equation

ax + by = c

in slope-intercept form.

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

This allows the identification of the slope of the line in terms of the coefficients

a

and

b

.

2

Set the slope expression equal to the given slope of

\frac{2}{3}

and solve for

b

in terms of

a

.

b = -1.5a

The slope of the line is

-\frac{a}{b}

, so

-\frac{a}{b} = \frac{2}{3} \implies 2b = -3a \implies b = -1.5a

.

3

Substitute

b = -1.5a

into the given equation

a + b = 5

and solve for

a

, then find

b

.

a = -10

and

b = 15

Substituting gives

a - 1.5a = 5 \implies -0.5a = 5 \implies a = -10

. Substituting

a = -10

back into the relationship gives

b = -1.5(-10) = 15

.

4

Substitute

a = -10

,

b = 15

, and the point

(6, 5)

into the equation

ax + by = c

and solve for

c

.

c = 15

(-10)(6) + (15)(5) = c \implies -60 + 75 = 15 \implies c = 15

.

Key Concept

Linear Equations in Two Variables

Question 239Question →

A hiker begins a climb at an elevation of $1,200$ feet above sea level and climbs at a constant rate of $350$ feet per hour. The hiker's elevation, $E$ , in feet, $t$ hours after beginning the climb is given by the equation $E = 350t + 1,200$ . Which of the following is the best interpretation of $350$ in this context?

The starting elevation, in feet, of the hiker

The increase in the hiker's elevation, in feet, for each hour of climbing

The number of hours it takes the hiker to climb one foot

The total elevation, in feet, of the hiker after 350 hours of climbing

Show answer & explanation

Answer: The increase in the hiker's elevation, in feet, for each hour of climbing

Answer

The increase in the hiker's elevation, in feet, for each hour of climbing

The coefficient of

t

, which is

350

, represents the rate of change of the hiker's elevation with respect to time. Since elevation is measured in feet and time is measured in hours, this rate is

350

feet per hour. Therefore, the value

350

represents the increase in the hiker's elevation, in feet, for each hour of climbing.

Step-by-Step Solution

1

Identify the structure of the linear equation.

The equation

E = 350t + 1,200

is in the slope-intercept form

y = mx + b

, where

m = 350

is the slope and

b = 1,200

is the

y

-intercept.

Linear equations in context have constant rates of change (slope) and starting values (

y

-intercept).

2

Determine the meaning of the slope in context.

The slope

350

represents the change in the dependent variable

E

(elevation in feet) per unit change in the independent variable

t

(time in hours).

The unit of the slope is the unit of the dependent variable divided by the unit of the independent variable, which is feet per hour.

3

Interpret the positive sign of the slope.

Since

350

is positive, the elevation increases by

350

feet for each hour of climbing.

A positive slope indicates a constant increase over time.

Key Concept

Interpreting slope in a linear context

Question 240Question →

If the solution to the inequality $a(2 - 3x) - 4(x + 3) \ge 12$ , where $a$ is a constant, is $x \le -1$ , what is the value of $a$ ?

Show answer & explanation

Answer: 4

Answer

The value of

a

is

4

.

To solve the inequality

a(2 - 3x) - 4(x + 3) \ge 12

, we first expand it to get

2a - 3ax - 4x - 12 \ge 12

. Grouping the

x

terms gives

(-3a - 4)x \ge 24 - 2a

. Since the inequality's solution is

x \le -1

, the direction of the inequality must flip, which means the coefficient of

x

, namely

-3a - 4

, must be negative. Dividing both sides by this coefficient gives the boundary value of the inequality as

\frac{24 - 2a}{-3a - 4}

. Setting this boundary equal to

-1

yields

24 - 2a = 3a + 4

, which simplifies to

5a = 20

, or

a = 4

. Since

a = 4

makes the coefficient

-3(4) - 4 = -16

negative, the solution holds.

Step-by-Step Solution

1

Expand the inequality using the distributive property.

2a - 3ax - 4x - 12 \ge 12

This allows us to separate and group the terms containing the variable

x

and the constant terms.

2

Group like terms and isolate the variable terms on the left-hand side.

(-3a - 4)x \ge 24 - 2a

By combining the coefficients of

x

and adding

12 - 2a

to both sides, we prepare the inequality to solve for

x

.

3

Determine the effect of dividing by the variable's coefficient.

Since the given solution is

x \le -1

, the inequality sign flipped from

\ge

to

\le

. Therefore, the coefficient

-3a - 4

must be negative.

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

4

Set the boundary value of the solution equal to

-1

and solve for

a

.

a = 4

Setting the boundary of the inequality

\frac{24 - 2a}{-3a - 4}

equal to

-1

allows us to find the specific constant

a

that produces this solution set.

Key Concept

Solving linear inequalities in one variable involving parameters and sign flips.

Algebra

Answer

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