Algebra Practice Questions — SAT — Page 11

Question 201Question →

In the $xy$ -plane, the system of linear equations below has infinitely many solutions, where $a$ , $b$ , and $c$ are constants and $c > 0$ :

\begin{aligned} 3x - 4y &= 10 \\ ax + by &= c \end{aligned}

If the graph of the second equation in the system passes through the point $(a, b)$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

Since the system has infinitely many solutions, the two equations are equivalent. This means the coefficients are proportional, so we can write

a = 3k

,

b = -4k

, and

c = 10k

for some constant

k

. Because the line passes through

(a, b)

, substituting these coordinates into the second equation gives

a^2 + b^2 = c

. Substituting the expressions in terms of

k

results in

(3k)^2 + (-4k)^2 = 10k

, which simplifies to

25k^2 = 10k

. Since

c > 0

, we have

k > 0

, and dividing by

k

gives

25k = 10

, so

k = 0.4

. Finally,

c = 10(0.4) = 4

.

Step-by-Step Solution

1

Set up a proportionality constant to relate the coefficients of the two equations.

a = 3k

,

b = -4k

, and

c = 10k

for a constant

k

.

Since the system has infinitely many solutions, the equations represent the same line, meaning their coefficients and constants must be proportional.

2

Substitute the point

(a, b)

into the second equation

ax + by = c

.

a^2 + b^2 = c

The graph of the equation passes through the point

(a, b)

, so the coordinates must satisfy the equation.

3

Substitute the parametric expressions of

a

,

b

, and

c

into the equation

a^2 + b^2 = c

.

25k^2 = 10k

This allows us to solve for the parameter

k

using a single variable quadratic equation.

4

Solve the equation

25k^2 = 10k

for

k

, given that

k > 0

.

k = 0.4

Since

c > 0

and

c = 10k

,

k

must be strictly positive, allowing us to divide both sides by

k

.

5

Compute the final value of

c

using the value of

k

.

c = 4

Substituting

k = 0.4

back into the expression

c = 10k

gives the value of

c

.

Key Concept

Systems of linear equations with infinitely many solutions and coordinate geometry constraints

Question 202Question →

If $\frac{2}{3}(6x - 9) - \frac{3}{4}(4x - 8) = \frac{1}{2}(x + 10)$ , what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10.

Distributing the fractions on the left-hand side of the equation simplifies the expression to

x

. Setting this equal to the right-hand side yields the equation

x = \frac{1}{2}(x + 10)

. Multiplying both sides by 2 gives

2x = x + 10

, and subtracting

x

from both sides gives the correct value of 10.

Step-by-Step Solution

1

Distribute the fraction

\frac{2}{3}

to the terms inside the first parentheses,

(6x - 9)

, and distribute the fraction

-\frac{3}{4}

to the terms inside the second parentheses,

(4x - 8)

.

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

Applying the distributive property removes the parentheses so that like terms can be combined.

2

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

x = \frac{1}{2}(x + 10)

Combining

4x

and

-3x

yields

x

, while

-6

and

6

cancel each other out.

3

Multiply both sides of the equation by 2 to eliminate the fraction.

2x = x + 10

Multiplying by the denominator simplifies the equation by removing the fraction.

4

Subtract

x

from both sides of the equation to isolate the variable.

x = 10

Subtracting

x

isolates the variable

x

on the left side of the equation.

Key Concept

Solving linear equations in one variable by applying the distributive property and combining like terms.

Question 203Question →

If $5(2a - 3) - 3(4a - 1) = -(a - 8)$ , what is the value of $a$ ?

-10

-4

-20

-16

Show answer & explanation

Answer: -20

Answer

-20

Distributing the constants on both sides of the equation

5(2a - 3) - 3(4a - 1) = -(a - 8)

yields

10a - 15 - 12a + 3 = -a + 8

. Combining like terms on the left side gives

-2a - 12 = -a + 8

. Adding

2a

to both sides results in

-12 = a + 8

. Subtracting

8

from both sides gives the correct value of

a

, which is

-20

.

Step-by-Step Solution

1

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

10a - 15 - 12a + 3 = -a + 8

To eliminate parentheses and allow terms to be combined, apply the distributive property:

5(2a - 3) = 10a - 15

,

-3(4a - 1) = -12a + 3

, and

-(a - 8) = -a + 8

.

2

Combine like terms on the left side of the equation.

-2a - 12 = -a + 8

Simplify the left side by combining the variable terms (

10a - 12a = -2a

) and the constant terms (

-15 + 3 = -12

).

3

Isolate the variable

a

on one side of the equation.

a = -20

Add

2a

to both sides of the equation to get

-12 = a + 8

, then subtract

8

from both sides to find that

a = -20

.

Key Concept

Solving linear equations in one variable using the distributive property, combining like terms, and isolating the variable.

Estimated Time:1m 30s

Question 204Question →

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

\begin{aligned} 3x - 2y &= 7 \\ kx + 3y &= 18 \end{aligned}

If the system has a solution $(x, y)$ such that $x + y = 4$ , what is the value of $k$ ?

3

5

9

1

Show answer & explanation

Answer: 5

Answer

5

The correct answer is

5

. To find the value of

k

, we first express

x

in terms of

y

using the given constraint equation:

x = 4 - y

. Next, we substitute this expression into the first equation of the system:

3(4 - y) - 2y = 7

. Distributing and combining like terms yields

12 - 5y = 7

. Solving for

y

gives

y = 1

. Substituting this back into the constraint equation gives

x = 4 - 1 = 3

. Finally, we substitute the solution point

(3, 1)

into the second equation:

k(3) + 3(1) = 18 \implies 3k + 3 = 18 \implies 3k = 15 \implies k = 5

.

Step-by-Step Solution

1

Express one variable in terms of the other using the given constraint equation

x + y = 4

.

x = 4 - y

This allows us to substitute the expression for

x

into the first equation to reduce it to a single variable.

2

Substitute

x = 4 - y

into the first equation

3x - 2y = 7

and solve for

y

.

3(4-y) - 2y = 7 \implies 12 - 3y - 2y = 7 \implies 12 - 5y = 7 \implies -5y = -5 \implies y = 1

By solving this linear equation, we find the unique y-coordinate of the system's solution.

3

Substitute the value of

y = 1

back into the constraint equation to find

x

.

x = 4 - 1 = 3

This determines the x-coordinate of the solution point, giving us the full solution

(3, 1)

.

4

Substitute the point

(3, 1)

into the second equation

kx + 3y = 18

to solve for the constant

k

.

k(3) + 3(1) = 18 \implies 3k + 3 = 18 \implies 3k = 15 \implies k = 5

Since the solution must satisfy all equations in the system, we can solve for

k

using the coordinates we found.

Key Concept

Solving systems of linear equations under linear constraints using algebraic substitution.

Estimated Time:2m 0s

Question 205Question →

Consider the system of equations below:

\begin{aligned} \frac{1}{3}x + \frac{1}{4}y &= 5 \\ x - y &= 8 \end{aligned}

If

(x, y)

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

y

?

Show answer & explanation

Answer: 4

Answer

The value of

y

is

4

.

Substituting

x = y + 8

from the second equation into the first equation yields

\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

. Multiplying the entire equation by

12

to clear denominators gives

4(y + 8) + 3y = 60

, which simplifies to

7y + 32 = 60

. Solving for

y

gives

7y = 28

, or

y = 4

.

Step-by-Step Solution

1

Express

x

in terms of

y

from the second equation.

x = y + 8

This allows for substitution into the first equation to solve for

y

directly.

2

Substitute

x = y + 8

into the first equation.

_\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

This reduces the system to a single-variable linear equation in terms of

y

.

3

Multiply the entire equation by the least common multiple of the denominators, which is

12

.

4(y + 8) + 3y = 60

This clears the fractional coefficients and simplifies the arithmetic.

4

Distribute the

4

and combine like terms.

7y + 32 = 60

Simplifies the equation to prepare for isolating the variable

y

.

5

Isolate the variable term by subtracting

32

from both sides, then dividing by

7

.

y = 4

This gives the final value of

y

that satisfies the system.

Key Concept

Solving systems of linear equations using substitution and clearing fractional coefficients

Question 206Question →

A state department of transportation models the relationship between the age of a highway, in years, and its road quality index on a scale from $0$ to $100$ . The index decreases at a constant rate with respect to the age of the highway. At age $4$ years, the highway has a road quality index of $82$ . At age $12$ years, the highway has a road quality index of $66$ . According to the model, after how many years from its construction will the highway's road quality index reach $50$ ?

Show answer & explanation

Answer: 20

Answer

According to the model, the highway's road quality index will reach

50

after

20

years.

Since the relationship is linear, the rate of change is constant. Using the points

(4, 82)

and

(12, 66)

, the slope is

\frac{66 - 82}{12 - 4} = -2

. The linear function can be written in point-slope form as

I - 82 = -2(t - 4)

, which simplifies to

I = -2t + 90

. Setting

I = 50

yields the equation

50 = -2t + 90

. Subtracting

90

from both sides gives

-40 = -2t

, and dividing by

-2

results in

t = 20

. Therefore, the index reaches

50

after

20

years.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the linear function.

The slope is

-2

.

Since the index decreases at a constant rate, the relationship is linear. The slope is calculated as the change in the road quality index divided by the change in the highway's age:

\frac{66 - 82}{12 - 4} = \frac{-16}{8} = -2

.

2

Determine the linear equation relating the road quality index,

I

, and the age,

t

.

I = -2t + 90

Using the point-slope formula with the point

(4, 82)

and slope

-2

:

I - 82 = -2(t - 4) \Rightarrow I - 82 = -2t + 8 \Rightarrow I = -2t + 90

.

3

Solve for the age,

t

, when the index

I

is

50

.

t = 20

Substitute

50

for

I

in the equation:

50 = -2t + 90

. Subtract

90

from both sides to get

-40 = -2t

. Divide by

-2

to find

t = 20

.

Key Concept

Writing and solving linear equations from two points

Question 207Question →

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

y \leq -x + 7

y \geq 2x - 5

y \geq 0

What is the maximum possible value of $x$ ?

3

4

7

2.5

Show answer & explanation

Answer: 4

Answer

The maximum possible value of

x

is 4.

To find the maximum possible value of

x

in the solution set, we analyze the boundaries of the system. The inequality

y \leq -x + 7

can be rewritten as

x \leq 7 - y

. The inequality

y \geq 2x - 5

can be rewritten as

x \leq 0.5y + 2.5

. For any point in the solution set,

x

must be less than or equal to both

7 - y

and

0.5y + 2.5

. Since

y \geq 0

, the maximum value of

x

will occur where these two upper bounds are equal:

7 - y = 0.5y + 2.5

, which simplifies to

1.5y = 4.5

, or

y = 3

. Substituting

y = 3

back into either equation gives

x = 4

. For any

y > 3

,

x \leq 7 - y < 4

. For any

y < 3

,

x \leq 0.5y + 2.5 < 4

. Thus, the maximum possible value of

x

is 4.

Step-by-Step Solution

1

Express the boundaries of

x

in terms of

y

from the given inequalities.

x \leq 7 - y

and

x \leq 0.5y + 2.5

To find the upper limits on

x

, we solve each inequality for

x

.

2

Find the intersection point of the two boundary lines by setting the expressions equal to each other.

7 - y = 0.5y + 2.5 \implies 1.5y = 4.5 \implies y = 3

. Substituting

y = 3

gives

x = 4

.

The maximum value of

x

occurs at the intersection of the two boundary constraints since one boundary increases with

y

and the other decreases with

y

.

3

Verify that the intersection point

(4, 3)

satisfies all three inequalities.

3 \leq -4 + 7

(true),

3 \geq 2(4) - 5

(true), and

3 \geq 0

(true).

Ensuring the point lies within the solution set confirms it is a valid maximum.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 208Question →

A shipping company uses a cargo plane to transport two types of cargo containers: Type A and Type B. Let $x$ represent the number of Type A containers, and let $y$ represent the number of Type B containers. The constraints on the shipment are represented by the system of inequalities below:

\begin{aligned} x + y &\leq 45 \\ 3x + 5y &\leq 165 \end{aligned}

If the plane must carry at least $10$ Type A containers, what is the maximum number of Type B containers the plane can transport?

25

27

35

39

Show answer & explanation

Answer: 27

Answer

27

The correct answer of 27 is found by substituting the minimum value of

x

, which is 10, into the second inequality constraint. This yields

3(10) + 5y \leq 165

, which simplifies to

5y \leq 135

, or

y \leq 27

. Since the coordinate pair (10, 27) also satisfies the first inequality constraint (

10 + 27 = 37 \leq 45

), 27 is the maximum possible value for

y

.

Step-by-Step Solution

1

Identify the given constraints and the variable to maximize.

We are given the system of inequalities

x + y \leq 45

and

3x + 5y \leq 165

, with

x \geq 10

, and we want to find the maximum possible value of

y

.

This establishes the mathematical model and boundaries for the problem.

2

Express

y

in terms of

x

for both inequalities to see which constraint is more restrictive.

From the first inequality:

y \leq 45 - x

. From the second inequality:

3x + 5y \leq 165 \implies 5y \leq 165 - 3x \implies y \leq 33 - 0.6x

.

This allows us to analyze

y

as a function of

x

under both constraints.

3

Substitute the minimum possible value of

x

, which is 10, into both boundary expressions for

y

.

Under the first constraint:

y \leq 45 - 10 = 35

. Under the second constraint:

y \leq 33 - 0.6(10) = 27

.

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

y

will occur when

x

is at its minimum value of 10.

4

Determine the most restrictive upper bound for

y

when

x = 10

.

The value of

y

must satisfy both

y \leq 35

and

y \leq 27

, so the maximum possible value is 27.

A solution to a system of inequalities must satisfy all inequalities in the system simultaneously.

Key Concept

Solving systems of linear inequalities in context by evaluating boundary conditions and identifying constraints.

Question 209Question →

In the $xy$ -plane, a point $(x, y)$ is in the solution set of the system of inequalities below:

x - 2y \leq -6

x + y \leq 11

y \leq 5

What is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The maximum possible value of

x

is

4

.

The maximum possible value of

x

is

4

. For all points in the solution set where

y \leq 5

, the active upper bound is

x \leq 2y - 6

. Since

2y - 6

increases as

y

increases, the maximum value of

x

occurs at the maximum boundary

y = 5

, giving

x = 2(5) - 6 = 4

.

Step-by-Step Solution

1

Express

x

in terms of

y

using the first two inequalities.

x \leq 2y - 6

and

x \leq 11 - y

To isolate the variable

x

and analyze how its upper bound is constrained by

y

.

2

Apply the third inequality constraint,

y \leq 5

, to find the limits on these upper bounds.

2y - 6 \leq 4

and

11 - y \geq 6

Since

y

cannot exceed

5

, the value of

2y - 6

is maximized when

y = 5

, and

11 - y

is minimized when

y = 5

.

3

Compare the two upper bounds to determine the active constraint for the domain

y \leq 5

.

Since

2y - 6 \leq 4

and

11 - y \geq 6

, the active bound is

x \leq 2y - 6

.

For any point to satisfy the system,

x

must be less than or equal to both bounds, meaning it is restricted by the smaller of the two bounds.

4

Calculate the maximum value of

x

at the boundary point.

x = 4

at the point

(4, 5)

The function

2y - 6

is increasing with respect to

y

, so its maximum value occurs at the largest possible value of

y

, which is

5

.

Key Concept

Finding the maximum value of a coordinate within a system of linear inequalities.

Question 210Question →

In the $xy$ -plane, a line with a positive slope $m$ passes through the points $(2, 5)$ and $(m, 13)$ . Which of the following is the $y$ -intercept of this line?

(0, 3)

(0, 9)

(0, -3)

(0, 1)

Show answer & explanation

Answer: $(0, -3)$

Answer

(0, -3)

The correct answer is the point

(0, -3)

. The slope of the line passing through

(2, 5)

and

(m, 13)

is expressed as

m = \frac{13 - 5}{m - 2} = \frac{8}{m - 2}

. Multiplying by

m - 2

yields the quadratic equation

m^2 - 2m - 8 = 0

. Factoring this equation gives

(m - 4)(m + 2) = 0

. Since the slope

m

is given as positive, we select

m = 4

. Using the point-slope form with

m = 4

and the point

(2, 5)

yields the equation

y - 5 = 4(x - 2)

, which simplifies to

y = 4x - 3

. The

y

-intercept of this line is found by setting

x = 0

, giving

y = -3

, or the coordinate point

(0, -3)

.

Step-by-Step Solution

1

Use the slope formula to express the slope of the line passing through

(2, 5)

and

(m, 13)

in terms of

m

.

m = \frac{13 - 5}{m - 2} \implies m = \frac{8}{m - 2}

The slope of a line is defined as the ratio of the change in

y

to the change in

x

between any two points on the line.

2

Solve the equation

m = \frac{8}{m - 2}

for

m

by converting it into a quadratic equation.

m(m - 2) = 8 \implies m^2 - 2m - 8 = 0 \implies (m - 4)(m + 2) = 0

, which gives

m = 4

or

m = -2

. Since the slope is positive,

m = 4

.

Multiplying both sides by the denominator clears the fraction and allows us to solve for the unknown parameter

m

using factoring.

3

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

m = 4

and the point

(2, 5)

.

y - 5 = 4(x - 2) \implies y = 4x - 8 + 5 \implies y = 4x - 3

The point-slope form provides a direct way to write the linear equation using a known point and the calculated slope.

4

Find the

y

-intercept by evaluating the line equation at

x = 0

.

y = 4(0) - 3 \implies y = -3

, which corresponds to the point

(0, -3)

.

The

y

-intercept of any line is the point where the line crosses the

y

-axis, which occurs at

x = 0

.

Key Concept

Linear Equations in Two Variables

Alternative Method

Alternatively, once the slope

m = 4

is found, we can write the equation of the line in slope-intercept form

y = 4x + b

. Substituting the coordinates of the point

(2, 5)

into the equation gives

5 = 4(2) + b \implies 5 = 8 + b

. Solving for

b

gives

b = -3

. Since

b

is the

y

-intercept, the point is

(0, -3)

.

Estimated Time:2m 30s

Question 211Question →

In the $xy$ -plane, a point $(x, y)$ lies in the solution set of the system of inequalities below.

$y \leq -x + 8$
$y \leq 2x + 2$

What is the maximum possible value of $y$ ?

Show answer & explanation

Answer: 6

Answer

The maximum possible value of

y

is 6.

The solution set to the system of inequalities is the region in the coordinate plane that lies on or below both boundary lines,

y = -x + 8

and

y = 2x + 2

. The maximum

y

-value in this region occurs at the intersection of the two lines. Solving the system of equations by setting

-x + 8 = 2x + 2

gives

3x = 6

, which simplifies to

x = 2

. Substituting

x = 2

back into either equation yields

y = 6

. Therefore, the maximum possible value of

y

is 6.

Step-by-Step Solution

1

Set the two boundary equations equal to each other to find the

x

-coordinate of the intersection point.

-x + 8 = 2x + 2 \implies 3x = 6 \implies x = 2

Since the solution region is bounded from above by both inequalities, the maximum value of

y

must occur at the intersection of the two boundary lines.

2

Substitute the

x

-value back into one of the equations to find the corresponding

y

-value.

y = 2(2) + 2 = 6

This determines the

y

-coordinate of the intersection point, which represents the maximum height of the shaded region.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:45s

Question 2120% of students answered this correctlyQuestion →

In the system of linear equations below, $c$ is a constant.

\begin{aligned} 2x + 3y &= 12 \\ cx - y &= 6 \end{aligned}

If the system has a unique solution $(x, y)$ such that $x > 0$ and $y < 0$ , which of the following could be the value of $c$ ?

-2

-1

\frac{1}{2}

2

Show answer & explanation

Answer: $\frac{1}{2}$

Answer

The option containing the value

\frac{1}{2}

is correct.

The correct answer is the option containing the value

\frac{1}{2}

. Solving the system of linear equations in terms of the constant

c

gives

x = \frac{30}{2+3c}

and

y = \frac{12c-12}{2+3c}

. For the solution

(x, y)

to lie in the fourth quadrant, we require

x > 0

and

y < 0

. The condition

x > 0

is satisfied when

2+3c > 0

, which simplifies to

c > -\frac{2}{3}

. Using this result, the condition

y < 0

requires the numerator of

y

to be negative, so

12c - 12 < 0

, which simplifies to

c < 1

. Combining these inequalities yields the interval

-\frac{2}{3} < c < 1

. The only value among the given options that falls within this interval is

\frac{1}{2}

.

Step-by-Step Solution

1

Express

y

in terms of

x

and the constant

c

using the second equation.

y = cx - 6

This allows for substitution into the first equation to solve for

x

.

2

Substitute the expression for

y

into the first equation and solve for

x

.

2x + 3(cx - 6) = 12 \implies (2 + 3c)x - 18 = 12 \implies (2 + 3c)x = 30 \implies x = \frac{30}{2 + 3c}

This isolates the variable

x

in terms of the constant

c

.

3

Substitute the expression for

x

back into the equation for

y

to express

y

in terms of

c

.

y = c\left(\frac{30}{2 + 3c}\right) - 6 = \frac{30c - 6(2 + 3c)}{2 + 3c} = \frac{12c - 12}{2 + 3c}

This isolates the variable

y

in terms of the constant

c

.

4

Apply the condition

x > 0

to find the constraint on

c

.

Since

x = \frac{30}{2 + 3c} > 0

and the numerator is positive, the denominator must also be positive:

2 + 3c > 0 \implies c > -\frac{2}{3}

.

This determines the lower bound for the constant

c

.

5

Apply the condition

y < 0

to find the constraint on

c

.

Since

y = \frac{12c - 12}{2 + 3c} < 0

and the denominator

2 + 3c

is positive, the numerator must be negative:

12c - 12 < 0 \implies c < 1

.

This determines the upper bound for the constant

c

.

6

Combine the inequalities to find the complete range for

c

and identify the matching option.

-\frac{2}{3} < c < 1

. The only value among the options that lies in this interval is

\frac{1}{2}

.

This identifies the correct option based on the mathematical constraints.

Key Concept

Solving systems of linear equations with parameters and applying quadrant boundary constraints.

Question 213Question →

Consider the system of inequalities shown below:

\begin{aligned} y &\geq 2x + 1 \\ y &< -x + 6 \end{aligned}

Which coordinate pair $(x, y)$ is a solution to the system?

(2, 3)

(1, 4)

(1, 5)

(2, 5)

Show answer & explanation

Answer: $(1, 4)$

Answer

The coordinate pair

(1, 4)

is a solution to the system of inequalities.

The coordinate pair

(1, 4)

is the correct answer because substituting

x = 1

and

y = 4

into both inequalities yields true statements:

4 \geq 2(1) + 1 \Rightarrow 4 \geq 3

is true, and

4 < -1 + 6 \Rightarrow 4 < 5

is true.

Step-by-Step Solution

1

Substitute the coordinates of the correct pair,

(1, 4)

, into the first inequality,

y \geq 2x + 1

.

4 \geq 2(1) + 1 \Rightarrow 4 \geq 3

, which is true.

To be a solution to the system, the point must satisfy the first inequality.

2

Substitute the same coordinates,

(1, 4)

, into the second inequality,

y < -x + 6

.

4 < -1 + 6 \Rightarrow 4 < 5

, which is true.

The point must also satisfy the second inequality to be a solution to the entire system.

3

Conclude that since

(1, 4)

satisfies both inequalities, it is a solution to the system.

The point

(1, 4)

is in the solution set.

A coordinate pair is a solution to a system of inequalities if and only if it satisfies every inequality in the system.

Key Concept

Testing coordinate pairs in a system of linear inequalities by substitution.

Estimated Time:45s

Question 214Question →

A nutritionist is designing a diet plan using two food types, Food A and Food B. The table below shows the amount of protein and carbohydrates, in grams, per gram of each food type:

Food Type	Protein (g)	Carbohydrates (g)
Food A	$0.12$	$0.08$
Food B	$0.18$	$0.05$

The nutritionist wants to combine the two foods to obtain a mixture that has a total mass of $250$ grams and contains exactly $36$ grams of protein. How many grams of carbohydrates will be in this mixture?

Show answer & explanation

Answer: 17

Answer

The total amount of carbohydrates in the mixture is 17 grams.

To find the total amount of carbohydrates in the mixture, we first define

x

as the mass in grams of Food A and

y

as the mass in grams of Food B. Since the total mass of the mixture is

250

grams, we write the equation

x + y = 250

. Since the total protein in the mixture must be

36

grams, and Food A and Food B contain

0.12

grams and

0.18

grams of protein per gram respectively, we write the equation

0.12x + 0.18y = 36

. Multiplying the first equation by

12

yields

12x + 12y = 3000

, and multiplying the second equation by

100

yields

12x + 18y = 3600

. Subtracting the first new equation from the second gives

6y = 600

, which simplifies to

y = 100

. Substituting this back into the first equation gives

x + 100 = 250

, so

x = 150

. Using these values, the total amount of carbohydrates is

0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

Step-by-Step Solution

1

Define variables for the mass of each food type.

Let

x

be the mass, in grams, of Food A, and let

y

be the mass, in grams, of Food B.

To represent the unknown quantities in the system of equations.

2

Set up the system of linear equations.

x + y = 250

(total mass equation) and

0.12x + 0.18y = 36

(total protein equation).

The mixture has a total mass of

250

grams and must contain exactly

36

grams of protein.

3

Solve the system of equations for

x

and

y

.

Multiply the first equation by

12

to get

12x + 12y = 3000

. Multiply the second equation by

100

to get

12x + 18y = 3600

. Subtract the first modified equation from the second to get

6y = 600

, which gives

y = 100

. Substitute

y = 100

back into

x + y = 250

to get

x = 150

.

To find the specific masses of Food A and Food B required for the mixture.

4

Calculate the total mass of carbohydrates in the mixture.

Total carbohydrates

= 0.08x + 0.05y = 0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

The question asks for the total grams of carbohydrates in the final mixture.

Key Concept

Systems of Linear Equations

Question 215Question →

An empty water tank is being filled by two hoses. Hose A fills the tank at a constant rate of 12 gallons per minute. Hose B fills the tank at a constant rate of 15 gallons per minute. Hose A is turned on first, and 5 minutes later, Hose B is also turned on. If both hoses run until the tank contains a total of 330 gallons of water, for how many minutes was Hose A running?

Show answer & explanation

Answer: 15

Answer

15

Let

m

be the number of minutes Hose A was running. Since Hose B was turned on 5 minutes after Hose A, Hose B was running for

m - 5

minutes. The volume of water contributed by Hose A is

12m

gallons, and the volume of water contributed by Hose B is

15(m - 5)

gallons. The sum of these volumes is 330 gallons, which gives the equation

12m + 15(m - 5) = 330

. Distributing 15 gives

12m + 15m - 75 = 330

. Combining like terms yields

27m - 75 = 330

. Adding 75 to both sides gives

27m = 405

. Dividing both sides by 27 yields

m = 15

. Thus, Hose A was running for 15 minutes.

Step-by-Step Solution

1

Define variables for the running time of each hose.

Let Hose A run for

m

minutes. Then Hose B runs for

m - 5

minutes.

Hose B started 5 minutes after Hose A, so it ran for 5 fewer minutes.

2

Set up a linear equation for the total gallons of water.

12m + 15(m - 5) = 330

The sum of the water from both hoses must equal 330 gallons.

3

Simplify and solve the linear equation.

27m = 405

, which gives

m = 15

.

Distribute, combine like terms, and isolate the variable

m

.

Key Concept

Setting up and solving a multi-step linear equation in one variable from a real-world context.

Question 216Question →

A public library is purchasing new books for its collection. The library plans to buy paperback books for $6 each and hardcover books for$ 20 each. The library has a budget of at most $1,200 for this purchase, wants to buy at most 120 total books, and must purchase at least 45 hardcover books. What is the maximum number of paperback books the library can purchase?

Show answer & explanation

Answer: 50

Answer

The maximum number of paperback books the library can purchase is 50.

The correct maximum number of paperback books is 50. By expressing the budget constraint as

p \le 200 - \frac{10}{3}h

, we see that the number of paperback books is maximized when the number of hardcover books,

h

, is minimized. Since the library must buy at least 45 hardcover books, we substitute

h = 45

into the inequality to get

p \le 50

. This combination also satisfies the total book constraint because

50 + 45 = 95

, which is less than or equal to 120.

Step-by-Step Solution

1

Set up the system of inequalities representing the library's constraints.

Let

p

be the number of paperbacks and

h

be the number of hardcovers. The constraints are

6p + 20h \le 1200

,

p + h \le 120

, and

h \ge 45

.

This translates the word problem into mathematical inequalities.

2

Isolate the variable

p

in the budget inequality to express it in terms of

h

.

p \le 200 - \frac{10}{3}h

This shows how the number of paperbacks depends on the number of hardcovers purchased.

3

Maximize

p

by using the minimum possible value of

h

.

p \le 200 - \frac{10}{3}(45) = 50

To maximize

p

, we must minimize

h

because buying more hardcovers decreases the remaining budget for paperbacks. The minimum value for

h

is 45.

4

Verify if the solution satisfies the remaining total books constraint.

50 + 45 = 95 \le 120

, which is true.

This ensures the solution is feasible and does not violate any other constraints.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 217Question →

For a constant $k$ , the graph of the linear equation $kx - 4y = 15$ in the $xy$ -plane contains the point $(3, 3)$ . What is the slope of the line?

Show answer & explanation

Answer: 2.25

Answer

2.25

Substituting the coordinates of the point

(3, 3)

into the given equation yields

k(3) - 4(3) = 15

, which simplifies to

3k - 12 = 15

. Adding

12

to both sides gives

3k = 27

, and dividing by

3

gives

k = 9

. Substituting

k = 9

back into the original equation gives

9x - 4y = 15

. To find the slope of the line, the equation can be written in slope-intercept form,

y = mx + b

, by isolating

y

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing all terms by

-4

yields

y = \frac{9}{4}x - \frac{15}{4}

, or

y = 2.25x - 3.75

. Therefore, the slope of the line is

\frac{9}{4}

, or

2.25

.

Step-by-Step Solution

1

Substitute the point

(3, 3)

into the equation

kx - 4y = 15

3k - 12 = 15

Since the graph of the equation contains the point

(3, 3)

, substituting

x = 3

and

y = 3

must satisfy the equation.

2

Solve for the constant

k

k = 9

Adding

12

to both sides of the equation gives

3k = 27

. Dividing both sides by

3

yields

k = 9

.

3

Substitute

k = 9

into the equation and rewrite it in slope-intercept form (

y = mx + b

)

y = \frac{9}{4}x - \frac{15}{4}

Substituting

k = 9

gives

9x - 4y = 15

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing both sides by

-4

isolates

y

.

4

Identify the slope from the slope-intercept form

Slope =

2.25

(or

\frac{9}{4}

)

In the slope-intercept form

y = mx + b

, the coefficient

m

represents the slope of the line.

Key Concept

Determining the slope of a line from a linear equation by first finding an unknown constant using a given point on the line.

Question 218Question →

A beverage company produces a fruit juice blend using apple juice and cranberry juice. Let $x$ represent the volume, in liters, of apple juice, and let $y$ represent the volume, in liters, of cranberry juice used in one batch of the blend. The production constraints for each batch are modeled by the system of inequalities below:

x + y \geq 100

y \geq \frac{1}{3}x

y \leq -\frac{1}{2}x + 75

What is the maximum possible volume, in liters, of cranberry juice that can be used in a single batch?

Show answer & explanation

Answer: 50

Answer

50

To find the maximum possible value of

y

, we examine the bounds of the feasible region. Rewriting the inequalities in terms of

x

gives

x \geq 100 - y

and

x \leq 150 - 2y

. Since

x

must satisfy both inequalities simultaneously, the lower bound must be less than or equal to the upper bound, meaning

100 - y \leq 150 - 2y

. Solving this inequality for

y

gives

y \leq 50

. The point

(50, 50)

also satisfies the remaining constraint

y \geq \frac{1}{3}x

, confirming that 50 is the maximum possible value.

Step-by-Step Solution

1

Isolate

x

in the first inequality:

x + y \geq 100

becomes

x \geq 100 - y

.

x \geq 100 - y

This establishes a lower bound for the volume of apple juice in terms of the volume of cranberry juice.

2

Isolate

x

in the third inequality:

y \leq -\frac{1}{2}x + 75

can be rewritten by subtracting 75 from both sides to get

y - 75 \leq -\frac{1}{2}x

, then multiplying by

-2

and reversing the inequality sign to get

x \leq 150 - 2y

.

x \leq 150 - 2y

This establishes an upper bound for the volume of apple juice in terms of the volume of cranberry juice.

3

Combine the lower and upper bounds of

x

:

100 - y \leq x \leq 150 - 2y

. For a valid solution to exist, the lower bound must be less than or equal to the upper bound.

100 - y \leq 150 - 2y

This allows us to solve directly for the constraint on the cranberry juice volume

y

.

4

Solve the inequality

100 - y \leq 150 - 2y

by adding

2y

to both sides and subtracting 100 from both sides.

y \leq 50

This determines that the maximum possible value for

y

is 50.

5

Substitute

y = 50

back into the compound inequality to find

x

:

100 - 50 \leq x \leq 150 - 2(50) \implies 50 \leq x \leq 50

, which means

x = 50

. Check if the point

(50, 50)

satisfies the remaining inequality

y \geq \frac{1}{3}x

:

50 \geq \frac{1}{3}(50) \approx 16.67

, which is true.

The point

(50, 50)

is a valid solution to the system.

This confirms that 50 is a achievable value within the system of inequalities.

Key Concept

Finding the boundary limits and optimization points within a system of linear inequalities in two variables.

Question 219Question →

A hot air balloon is at an altitude of $h$ meters. The balloon begins to descend at a constant rate. After $3$ minutes, the altitude of the balloon is $540$ meters. After $8$ minutes, the altitude of the balloon is $390$ meters. If the altitude of the balloon is modeled by a linear function of time, what was the initial altitude of the balloon, in meters?

Show answer & explanation

Answer: 630

Answer

The initial altitude of the balloon was 630 meters.

To find the initial altitude, we model the balloon's descent as a linear equation of the form

h(t) = mt + b

, where

h(t)

is the altitude at time

t

,

m

is the constant rate of change, and

b

is the initial altitude. Using the given points

(3, 540)

and

(8, 390)

, we find the slope

m = \frac{390 - 540}{8 - 3} = -30

meters per minute. Substituting

m = -30

and the point

(3, 540)

into

h(t) = mt + b

gives

540 = -30(3) + b

. Solving for

b

yields

b = 540 + 90 = 630

meters.

Step-by-Step Solution

1

Identify two data points from the problem context

The coordinate points are

(3, 540)

and

(8, 390)

To define the linear relationship, we need at least two coordinate points representing (time, altitude)

2

Calculate the slope (

m

) of the linear function

m = \frac{390 - 540}{8 - 3} = \frac{-150}{5} = -30

The rate of change represents the speed at which the balloon descends each minute

3

Solve for the vertical intercept (

b

) using the slope-intercept form

y = mx + b

540 = -30(3) + b \implies b = 630

The initial altitude corresponds to the altitude at time

t = 0

, which is the vertical intercept of the linear function

Key Concept

Linear Functions and Graphs

Question 220Question →

In the $xy$ -plane, the region $R$ consists of all points $(x, y)$ that satisfy the system of inequalities below:

\begin{aligned} 3x - 4y &\geq -12 \\ y &\geq 2x - 7 \\ x + 2y &\geq 2 \end{aligned}

If $(x, y)$ is a point in region $R$ , what is the maximum possible value of the expression $x - 2y$ ?

2.0

4.4

5.0

-5.2

Show answer & explanation

Answer: 4.4

Answer

4.4

The maximum value of a linear expression over a closed, bounded polygonal region must occur at one of the vertices of that region. By solving the systems of equations for each pair of boundary lines, the vertices of the triangular region are found to be

(8, 9)

,

(3.2, -0.6)

, and

(-1.6, 1.8)

. Evaluating the expression

x - 2y

at each vertex gives

-10

at

(8, 9)

,

4.4

at

(3.2, -0.6)

, and

-5.2

at

(-1.6, 1.8)

. Comparing these values,

4.4

is the maximum possible value.

Step-by-Step Solution

1

Find the intersection points of the boundary equations by solving the three systems of two linear equations.

System 1 (

3x - 4y = -12

and

y = 2x - 7

) yields

(8, 9)

.
System 2 (

y = 2x - 7

and

x + 2y = 2

) yields

(3.2, -0.6)

.
System 3 (

3x - 4y = -12

and

x + 2y = 2

) yields

(-1.6, 1.8)

.

The boundaries of the region are defined by these lines, so the vertices of the region occur at their intersections.

2

Verify that the region defined by the inequalities is the interior of the triangle formed by these vertices by testing a point inside, such as

(2, 2)

.

The point

(2, 2)

satisfies all three inequalities:

3(2) - 4(2) = -2 \geq -12

(True)

2 \geq 2(2) - 7 = -3

(True)

2 + 2(2) = 6 \geq 2

(True)

This confirms that the region is bounded and the vertices are the extreme points of the closed triangular region.

3

Evaluate the expression

x - 2y

at each of the three vertices.

At

(8, 9)

:

8 - 2(9) = -10

At

(3.2, -0.6)

:

3.2 - 2(-0.6) = 3.2 + 1.2 = 4.4

At

(-1.6, 1.8)

:

-1.6 - 2(1.8) = -1.6 - 3.6 = -5.2

A linear expression defined over a closed, bounded polygonal region reaches its maximum and minimum values at the vertices of that region.

4

Compare the evaluated values to identify the maximum value.

The maximum value is

4.4

.

Among the three candidate values (

-10

,

4.4

, and

-5.2

),

4.4

is the greatest.

Key Concept

Linear programming vertex method for systems of linear inequalities.

Estimated Time:3m 0s

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