Algebra Practice Questions — SAT — Page 5

Question 81Question →

If $3(x - 2) = -x + 14$ , what is the value of $x$ ?

2

4

5

10

Show answer & explanation

Answer: 5

Answer

5

The correct value is 5. Distributing the 3 on the left side of the equation

3(x - 2) = -x + 14

yields

3x - 6 = -x + 14

. Adding

x

to both sides gives

4x - 6 = 14

. Adding 6 to both sides gives

4x = 20

. Finally, dividing both sides by 4 yields

x = 5

.

Step-by-Step Solution

1

Distribute the 3 to the terms inside the parentheses on the left side of the equation.

3x - 6 = -x + 14

To simplify the equation and prepare to isolate the variable, the parentheses must be expanded.

2

Add

x

to both sides of the equation to group all variable terms on the left side.

4x - 6 = 14

Grouping the variable terms helps in isolating the variable.

3

Add 6 to both sides of the equation to isolate the variable term.

4x = 20

Moving the constant term to the other side isolates the variable term

4x

.

4

Divide both sides of the equation by 4 to find the value of

x

.

x = 5

Dividing by the coefficient of

x

gives the final value of the variable.

Key Concept

Linear Equations in One Variable

Estimated Time:45s

Question 82Question →

In the $xy$ -plane, line $l_1$ passes through the origin and has a positive slope $m$ , where $m > 1$ . Line $l_2$ is perpendicular to line $l_1$ and has a $y$ -intercept of $(0, 10)$ . The two lines intersect at point $P$ . If the distance from point $P$ to the $y$ -axis is $4$ , what is the value of $m$ ?

Show answer & explanation

Answer: 2

Answer

2

The equation of line

l_1

is

y = mx

and the equation of line

l_2

is

y = -\frac{1}{m}x + 10

. Setting these equal gives the

x

-coordinate of their intersection as

x = \frac{10m}{m^2 + 1}

. Since the distance from the intersection point to the

y

-axis is

4

, we have

\frac{10m}{m^2 + 1} = 4

. Solving this quadratic equation yields

m = 2

or

m = \frac{1}{2}

. Given that

m > 1

, the correct value is

2

.

Step-by-Step Solution

1

Write the equations of lines

l_1

and

l_2

.

Line

l_1

has a slope of

m

and passes through

(0,0)

, so its equation is

y = mx

. Line

l_2

is perpendicular to

l_1

, so its slope is

-\frac{1}{m}

. Since its

y

-intercept is

(0,10)

, its equation is

y = -\frac{1}{m}x + 10

.

Setting up the equations of the lines allows us to find their point of intersection.

2

Find the

x

-coordinate of the intersection point

P

.

Equating the two expressions for

y

gives

mx = -\frac{1}{m}x + 10

. Multiplying both sides by

m

yields

m^2 x = -x + 10m

, which simplifies to

(m^2 + 1)x = 10m

, or

x = \frac{10m}{m^2 + 1}

.

The intersection point

P

must satisfy both equations simultaneously.

3

Solve for

m

using the distance from

P

to the

y

-axis.

The distance from

P(x,y)

to the

y

-axis is given by

|x|

. Since

m > 1

,

x

is positive, so the distance is

\frac{10m}{m^2 + 1} = 4

. This simplifies to

10m = 4m^2 + 4

, or

4m^2 - 10m + 4 = 0

. Dividing by

2

gives

2m^2 - 5m + 2 = 0

. Factoring the quadratic yields

(2m - 1)(m - 2) = 0

, giving solutions of

m = \frac{1}{2}

and

m = 2

. Since

m > 1

, the slope of

l_1

must be

2

.

Applying the given distance constraint and slope condition determines the unique value of

m

.

Key Concept

The relationship between the equations of perpendicular lines and their point of intersection in the coordinate plane.

Question 83Question →

The graph of the linear function $f$ in the $xy$ -plane has a $y$ -intercept of $(0, b)$ and an $x$ -intercept of $(a, 0)$ , where $a$ and $b$ are nonzero constants. If $3a = -4b$ , which of the following is the slope of the graph of $f$ ?

-\frac{4}{3}

-\frac{3}{4}

\frac{3}{4}

\frac{4}{3}

Show answer & explanation

Answer: $\frac{3}{4}$

Answer

\frac{3}{4}

The slope of the line passing through

(0, b)

and

(a, 0)

is

m = \frac{0 - b}{a - 0} = -\frac{b}{a}

. Starting with the given equation

3a = -4b

, dividing both sides by

-4a

isolates the slope expression:

-\frac{b}{a} = \frac{3}{4}

. Therefore, the slope of the graph of

f

is

\frac{3}{4}

.

Step-by-Step Solution

1

Identify the coordinates of the intercepts and write the formula for the slope of a line.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(a, 0)

. The slope

m

is given by

m = \frac{0 - b}{a - 0} = -\frac{b}{a}

.

To find the slope, we express it in terms of the variables

a

and

b

using the standard slope formula.

2

Use the given equation to find the ratio

-\frac{b}{a}

.

Divide both sides of the equation

3a = -4b

by

a

to get

3 = -4\left(\frac{b}{a}\right)

. Then, divide both sides by

-4

to get

-\frac{3}{4} = \frac{b}{a}

, which means

-\frac{b}{a} = \frac{3}{4}

.

We isolate the expression for the slope, which is

-\frac{b}{a}

, using algebraic operations on the given equation.

3

Equate the slope expression to the calculated value.

Since

m = -\frac{b}{a}

and

-\frac{b}{a} = \frac{3}{4}

, the slope of the line is

\frac{3}{4}

.

This yields the final value of the slope.

Key Concept

Calculating the slope of a linear function using intercepts and algebraic substitution.

Question 84Question →

The table below shows several values of $x$ and their corresponding values of $f(x)$ for the linear function $f$ .

$x$	$f(x)$
$2$	$7$
$5$	$13$
$8$	$19$

What is the slope of the graph of $y = f(x)$ in the $xy$ -plane?

2

0.5

3

4

Show answer & explanation

Answer: 2

Answer

The slope of the graph of

y = f(x)

is 2.

The slope of a linear function represents the constant rate of change of the function values with respect to the input values. Using any two points from the table, such as

(2, 7)

and

(5, 13)

, the slope

m

can be calculated using the slope formula:

m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{13 - 7}{5 - 2} = \frac{6}{3} = 2

. Therefore, the slope of the graph of

y = f(x)

is

2

.

Step-by-Step Solution

1

Select two points from the given table.

Two points are

(2, 7)

and

(5, 13)

.

To find the slope of a linear function, we need to calculate the rate of change between any two points on the line.

2

Apply the slope formula

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

.

m = \frac{13 - 7}{5 - 2} = \frac{6}{3} = 2

Calculating the ratio of the vertical change (change in

f(x)

) to the horizontal change (change in

x

) yields the constant slope of the linear function.

Key Concept

Calculating the slope of a linear function from a table of values.

Estimated Time:1m 0s

Question 85Question →

If $3(2n + 4) = 42$ , what is the value of $n$ ?

Show answer & explanation

Answer: 5

Answer

5

To solve the linear equation

3(2n + 4) = 42

, first divide both sides of the equation by 3, which simplifies it to

2n + 4 = 14

. Next, subtract 4 from both sides of the equation to isolate the term with the variable, giving

2n = 10

. Finally, divide both sides of the equation by 2 to solve for the variable, which yields

n = 5

.

Step-by-Step Solution

1

Divide both sides of the equation by 3.

2n + 4 = 14

To simplify the equation by eliminating the outer constant multiplier.

2

Subtract 4 from both sides of the equation.

2n = 10

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

3

Divide both sides of the equation by 2.

n = 5

To find the value of

n

.

Key Concept

Solving a linear equation in one variable using inverse operations.

Estimated Time:45s

Question 86Question →

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $y = 2x - 7$ . If the region in the first quadrant bounded by the graph of $f$ , the $x$ -axis, and the $y$ -axis has an area of $36$ , what is the $x$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 12

Answer

The

x

-intercept of the graph of

f

is

12

.

The correct answer is

12

. Since the graph of

f

is perpendicular to the line

y = 2x - 7

, its slope is

-\frac{1}{2}

. The equation of the line is

f(x) = -\frac{1}{2}x + b

, which has a

y

-intercept of

(0, b)

and an

x

-intercept of

(2b, 0)

. In the first quadrant, these intercepts form a right triangle with the axes, having legs of length

b

and

2b

. The area of this triangle is

\frac{1}{2}(2b)(b) = b^2

. Setting the area equal to

36

gives

b^2 = 36

, so

b = 6

(since

b > 0

). The

x

-intercept is

2b = 2(6) = 12

.

Step-by-Step Solution

1

Find the slope of the perpendicular line

f

.

The slope of

f

is

-\frac{1}{2}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The given line has a slope of

2

, so the slope of

f

must be

-\frac{1}{2}

.

2

Express the

y

-intercept and

x

-intercept of

f

in terms of a single variable.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(2b, 0)

, where

b > 0

.

The equation of

f

can be written as

f(x) = -\frac{1}{2}x + b

. Setting

x = 0

gives the

y

-intercept

b

. Setting

f(x) = 0

and solving for

x

gives the

x

-intercept

2b

.

3

Set up the area equation for the triangle in the first quadrant.

The area is represented by

b^2 = 36

.

The region bounded by the graph of

f

and the coordinate axes in the first quadrant forms a right triangle with perpendicular sides of lengths

b

and

2b

. The area of this triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2b \times b = b^2

.

4

Solve for

b

and calculate the

x

-intercept.

The

x

-intercept is

12

.

Solving

b^2 = 36

with the condition

b > 0

yields

b = 6

. The

x

-intercept is

2b

, which equals

2(6) = 12

.

Key Concept

Using perpendicular slopes and intercepts of linear functions to analyze geometric areas in the coordinate plane.

Estimated Time:2m 0s

Question 87Question →

A commercial drone delivery service charges a flat rate of $15.50$ dollars per delivery plus a fuel surcharge of $1.80$ dollars per mile. During a promotional event, the service offers a $20\%$ discount off the fuel surcharge only. If the total cost for a specific delivery during this promotion was $27.02$ dollars, how many miles did the drone travel for this delivery?

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. The promotional fuel surcharge is reduced by

20\%

from the normal

1.80

dollars per mile, resulting in a rate of

1.80 \times (1 - 0.20) = 1.44

dollars per mile. The linear equation modeling the total cost for a delivery of

m

miles during the promotion is

15.50 + 1.44m = 27.02

. Subtracting

15.50

from both sides gives

1.44m = 11.52

. Dividing both sides by

1.44

yields the distance

m = 8

miles.

Step-by-Step Solution

1

Determine the discounted fuel surcharge per mile.

The discounted surcharge is

1.44

dollars per mile.

A

20\%

discount is applied to the original

1.80

dollars per mile fuel surcharge:

1.80 \times (1 - 0.20) = 1.44

.

2

Set up the linear equation representing the total promotional cost.

15.50 + 1.44m = 27.02

, where

m

represents the number of miles traveled.

The total cost is the sum of the flat rate of

15.50

dollars and the promotional fuel surcharge of

1.44

dollars per mile multiplied by the number of miles.

3

Subtract the flat rate from both sides of the equation.

1.44m = 11.52

To isolate the term with the variable

m

, subtract the flat rate of

15.50

from the total promotional cost of

27.02

.

4

Solve for the distance by division.

m = 8

Divide both sides of the equation by the promotional rate per mile,

1.44

, to find the number of miles.

Key Concept

Formulating and solving a multi-step linear equation in one variable with decimals and percentages to represent a real-world scenario.

Question 88Question →

A local municipal water utility charges a flat monthly service fee plus a constant rate for each hundred cubic feet (HCF) of water consumed. During one month, a household that consumed 12 HCF of water was charged 46.00 dollars. Another household that consumed 18 HCF of water was charged 61.00 dollars. What is the flat monthly service fee, in dollars, charged by the utility?

Show answer & explanation

Answer: 16

Answer

The flat monthly service fee is 16 dollars.

By modeling the relationship as a linear function

y = mx + b

, where

x

is the consumption in HCF and

y

is the total charge in dollars, we use the points

(12, 46)

and

(18, 61)

to find the slope

m = \frac{61 - 46}{18 - 12} = 2.5

. Substituting

m = 2.5

and the point

(12, 46)

back into the linear equation gives

46 = 2.5(12) + b

, which simplifies to

46 = 30 + b

. Solving for

b

gives 16, representing the flat service fee.

Step-by-Step Solution

1

Set up a system of points representing the linear relationship between water consumed and monthly charge.

Two coordinate points are established:

(12, 46)

and

(18, 61)

, where the first coordinate is the consumption in HCF and the second is the charge in dollars.

Since the utility charges a flat fee plus a constant rate, the relationship is linear and can be solved using coordinate points.

2

Find the constant rate per HCF (the slope of the line).

The slope

m

is calculated as

\frac{61 - 46}{18 - 12} = \frac{15}{6} = 2.5

dollars per HCF.

The slope represents the constant rate of change in total cost per HCF of water consumed.

3

Determine the flat monthly service fee (the y-intercept of the line).

Using the slope-intercept equation

y = mx + b

with point

(12, 46)

and

m = 2.5

yields

46 = 2.5(12) + b \implies 46 = 30 + b \implies b = 16

.

The y-intercept represents the cost when consumption is zero, which is the flat monthly service fee.

Key Concept

Finding the y-intercept of a linear function from two points.

Estimated Time:1m 30s

Question 89Question →

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

-4

\frac{8}{3}

-5

-\frac{14}{3}

Show answer & explanation

Answer: $-4$

Answer

-4

The correct answer is

-4

. Distributing the terms on the left side of the equation

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

gives

2x - 3 - 2x - 10 = 3x - 1

. Simplifying the left side yields

-13 = 3x - 1

. Adding

1

to both sides results in

-12 = 3x

. Finally, dividing both sides by

3

gives

x = -4

.

Step-by-Step Solution

1

Distribute the constants outside the parentheses on the left side of the equation.

2x - 3 - 2x - 10 = 3x - 1

Distributing

\frac{1}{2}

to

(4x - 6)

yields

2x - 3

, and distributing

-2

to

(x + 5)

yields

-2x - 10

.

2

Combine like terms on the left side of the equation.

-13 = 3x - 1

Combining

2x

and

-2x

results in

0

, and combining

-3

and

-10

results in

-13

.

3

Isolate the variable

x

by adding

1

to both sides and dividing by

3

.

x = -4

Adding

1

to both sides gives

-12 = 3x

. Dividing both sides by

3

yields

x = -4

.

Key Concept

Solving linear equations in one variable by distributing constants and combining like terms.

Question 90Question →

If $12 - 2(w - 1) = 4$ , what is the value of $w$ ?

-3

3

5

9

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. We solve the equation by first distributing the coefficient

-2

to

(w - 1)

, which yields

12 - 2w + 2 = 4

. Combining the constant terms gives

14 - 2w = 4

. Subtracting 14 from both sides results in

-2w = -10

. Finally, dividing by

-2

isolates the variable, giving

w = 5

.

Step-by-Step Solution

1

Distribute the negative coefficient

-2

to the terms inside the parentheses.

12 - 2w + 2 = 4

To eliminate the parentheses and simplify the expression.

2

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

14 - 2w = 4

To group like terms together.

3

Subtract 14 from both sides of the equation.

-2w = -10

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

4

Divide both sides of the equation by

-2

.

w = 5

To find the value of

w

.

Key Concept

Solving linear equations in one variable using distributive properties and basic operations.

Question 91Question →

A system of linear equations is shown below.

y = 2x - 7

3x - 2y = 11

What is the value of $x$ ?

8

-25

3

-3

Show answer & explanation

Answer: 3

Answer

The correct value of

x

is 3.

The correct value of

x

is 3, which is found by substituting

2x - 7

for

y

in the second equation and solving for

x

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

3x - 2(2x - 7) = 11

Since the first equation defines

y

in terms of

x

, substitution is a direct way to eliminate

y

and solve for

x

.

2

Distribute the

-2

to both terms inside the parentheses.

3x - 4x + 14 = 11

Applying the distributive property requires multiplying

-2

by both

2x

and

-7

, remembering that negative times negative is positive.

3

Combine the like terms on the left side of the equation.

-x + 14 = 11

Simplifying the expression by combining the

x

terms (

3x - 4x = -x

).

4

Isolate the variable term by subtracting 14 from both sides.

-x = -3

Using the subtraction property of equality to move the constant term to the right side.

5

Solve for

x

by multiplying or dividing both sides by

-1

.

x = 3

Finding the positive value of

x

.

Key Concept

Solving systems of linear equations using the substitution method.

Question 92Question →

In the equation below,

p

and

q

are constants.

\frac{2}{p}(6x - 9) - \frac{3}{q}(4x + 2) = -2x - 9

If the equation has infinitely many solutions for

x

, what is the value of

p + q

?

1

5

9

-5

Show answer & explanation

Answer: 5

Answer

The value of

p+q

is

5

.

For the equation to have infinitely many solutions for

x

, the coefficients of

x

on both sides must be equal, and the constant terms on both sides must also be equal. Expanding the left side of the equation gives

\left(\frac{12}{p} - \frac{12}{q}\right)x - \left(\frac{18}{p} + \frac{6}{q}\right) = -2x - 9

. Equating the coefficients of

x

yields

\frac{12}{p} - \frac{12}{q} = -2

, which simplifies to

\frac{6}{p} - \frac{6}{q} = -1

. Equating the constant terms yields

-\frac{18}{p} - \frac{6}{q} = -9

, which simplifies to

\frac{18}{p} + \frac{6}{q} = 9

. Adding these two simplified equations eliminates the term with

q

, giving

\frac{24}{p} = 8

, which means

p = 3

. Substituting

p = 3

back into the first equation yields

2 - \frac{6}{q} = -1

, which simplifies to

\frac{6}{q} = 3

, meaning

q = 2

. Thus, the value of

p + q

is

3 + 2 = 5

.

Step-by-Step Solution

1

Expand and group the terms on the left side of the equation.

\left(\frac{12}{p} - \frac{12}{q}\right)x - \left(\frac{18}{p} + \frac{6}{q}\right) = -2x - 9

To analyze the linear equation, we must group the coefficients of the variable

x

and the constant terms.

2

Set up a system of equations for

p

and

q

using the condition for infinitely many solutions.

\frac{12}{p} - \frac{12}{q} = -2

and

-\frac{18}{p} - \frac{6}{q} = -9

A linear equation in the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

.

3

Simplify the system of equations.

\frac{6}{p} - \frac{6}{q} = -1

(Equation 1) and

\frac{18}{p} + \frac{6}{q} = 9

(Equation 2)

Dividing Equation 1 by

2

and Equation 2 by

-1

simplifies the coefficients, making the system easier to solve.

4

Solve for

p

by adding Equation 1 and Equation 2.

\frac{24}{p} = 8 \implies p = 3

Adding the two equations eliminates the term containing

q

, allowing us to solve directly for

p

.

5

Substitute

p = 3

back into Equation 1 to solve for

q

.

\frac{6}{3} - \frac{6}{q} = -1 \implies 2 - \frac{6}{q} = -1 \implies \frac{6}{q} = 3 \implies q = 2

Using the value of

p

allows us to isolate and solve for

q

.

6

Calculate the sum of

p

and

q

.

p + q = 3 + 2 = 5

The question asks for the value of

p + q

.

Key Concept

Conditions for a linear equation in one variable to have infinitely many solutions, and solving systems of literal equations.

Question 93Question →

A scientist is measuring the temperature of a sample that is being heated at a constant rate. At a starting time of $0$ minutes, the temperature of the sample is $12^\circ\text{C}$ . After $8$ minutes of heating, the temperature of the sample is $44^\circ\text{C}$ . If the temperature of the sample increases linearly with time, what is the temperature of the sample, in degrees Celsius, after $15$ minutes of heating?

Show answer & explanation

Answer: 72

Answer

72

The rate of temperature change is

\frac{44 - 12}{8 - 0} = 4^\circ\text{C}

per minute. Using the slope-intercept form, the temperature

T

after

t

minutes can be represented by the function

T(t) = 4t + 12

. Substituting

15

for

t

yields

T(15) = 4(15) + 12 = 60 + 12 = 72

.

Step-by-Step Solution

1

Calculate the constant rate of temperature change (slope).

4 degrees Celsius per minute

To find how much the temperature increases each minute, divide the total change in temperature by the total change in time:

\frac{44 - 12}{8 - 0} = \frac{32}{8} = 4

.

2

Set up the linear function for temperature

T(t)

over time

t

.

T(t) = 4t + 12

The initial temperature at

t = 0

is

12^\circ\text{C}

, which represents the vertical intercept. The constant rate of change is

4^\circ\text{C}

per minute.

3

Find the temperature at

t = 15

minutes.

72

Substitute

15

for

t

in the equation:

T(15) = 4(15) + 12 = 60 + 12 = 72

.

Key Concept

Linear Functions and Rates of Change

Estimated Time:1m 0s

Question 94Question →

A line in the $xy$ -plane has a slope of $-3$ . The line passes through the point $(2, k)$ and has an $x$ -intercept of $(r, 0)$ , where $k$ and $r$ are constants. If $r + k = 10$ , what is the $y$ -intercept of the line?

(0, 4)

(0, 6)

(0, 10)

(0, 12)

Show answer & explanation

Answer: $(0, 12)$

Answer

The y-intercept of the line is

(0, 12)

.

The correct answer is

(0, 12)

. By using the formula for slope

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

with the points

(2, k)

and

(r, 0)

, we obtain

-3 = \frac{k - 0}{2 - r}

. Multiplying both sides by

2 - r

gives

k = -3(2 - r)

, which simplifies to

k = -6 + 3r

, or

3r - k = 6

. We can solve this alongside the given equation

r + k = 10

by adding the two equations:

(3r - k) + (r + k) = 6 + 10 \implies 4r = 16

, which yields

r = 4

. Substituting

r = 4

back into the sum equation gives

k = 6

. Now, using the x-intercept point

(4, 0)

and the slope of

-3

, the equation of the line in point-slope form is

y - 0 = -3(x - 4)

, which simplifies to

y = -3x + 12

. The y-intercept of this line is found by setting

x = 0

, yielding

(0, 12)

.

Step-by-Step Solution

1

Express the slope of the line using the coordinates of the two given points,

(2, k)

and the x-intercept

(r, 0)

.

Using the slope formula

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

, we set up the equation:

-3 = \frac{k - 0}{2 - r}

.

This connects the geometric concept of slope to the algebraic coordinates of the points on the line.

2

Simplify the slope equation to relate

k

and

r

.

Multiplying both sides by

2 - r

gives

k = -3(2 - r) \implies k = -6 + 3r

, which can be rewritten as

3r - k = 6

.

Simplifying the equation makes it easier to solve as part of a system of linear equations.

3

Solve the system of equations consisting of

3r - k = 6

and the given equation

r + k = 10

.

Adding the two equations together:

(3r - k) + (r + k) = 6 + 10 \implies 4r = 16 \implies r = 4

. Substituting

r = 4

into

r + k = 10

gives

4 + k = 10 \implies k = 6

.

Solving the system of equations determines the numerical values of the constants

r

and

k

.

4

Find the equation of the line using the slope

m = -3

and the x-intercept

(4, 0)

, then determine the y-intercept.

Using point-slope form:

y - 0 = -3(x - 4) \implies y = -3x + 12

. The y-intercept occurs when

x = 0

, giving

y = 12

, which corresponds to the point

(0, 12)

.

Writing the full linear equation allows us to find the y-intercept by evaluating the function at

x = 0

.

Key Concept

Writing linear equations from given points and slope, and solving a system of linear equations to identify intercepts.

Estimated Time:2m 0s

Question 95Question →

If $2x + 7 = 15$ , what is the value of $4x - 3$ ?

Show answer & explanation

Answer: 13

Answer

13

To find the value of the expression, first solve the linear equation for the variable. Subtracting 7 from both sides of the equation yields

2x = 8

. Dividing both sides of the equation by 2 yields

x = 4

. Substituting this value back into the expression yields

4(4) - 3 = 16 - 3 = 13

. Alternatively, since

2x = 8

, multiplying both sides by 2 gives

4x = 16

. Subtracting 3 from both sides of this relation yields

4x - 3 = 16 - 3 = 13

.

Step-by-Step Solution

1

Subtract 7 from both sides of the equation

2x = 8

To isolate the variable term on one side of the equation

2

Divide both sides by 2

x = 4

To solve for the variable

3

Substitute the value of the variable into the target expression

13

To find the final evaluated value requested by the question

Key Concept

Solving a linear equation in one variable and evaluating an expression

Question 96Question →

A chemist mixes two saline solutions. Solution A is $12\%$ salt by mass, and Solution B is $30\%$ salt by mass. The mass of Solution B used in the mixture is 20 grams more than $\frac{1}{3}$ of the mass of Solution A used. If the resulting mixture is $18\%$ salt by mass, what is the mass, in grams, of Solution A used?

Show answer & explanation

Answer: 120

Answer

The mass of Solution A used is 120 grams.

The correct answer is 120. By setting the mass of Solution A to

x

, the mass of Solution B is

\frac{1}{3}x + 20

. Equating the total salt content from both individual solutions to the total salt content of the final mixture gives the linear equation

0.12x + 0.30\left(\frac{1}{3}x + 20\right) = 0.18\left(x + \frac{1}{3}x + 20\right)

. Simplifying both sides yields

0.22x + 6 = 0.24x + 3.6

. Solving this equation gives

0.02x = 2.4

, which simplifies to

x = 120

.

Step-by-Step Solution

1

Define the variable

x

as the mass, in grams, of Solution A used in the mixture, and express the mass of Solution B in terms of

x

.

Mass of Solution A =

x

grams; Mass of Solution B =

\frac{1}{3}x + 20

grams.

To set up expressions representing the mass of each solution in the mixture.

2

Calculate the mass of salt contributed by each solution and write an expression for the total mass of salt.

Salt from Solution A =

0.12x

grams; Salt from Solution B =

0.30\left(\frac{1}{3}x + 20\right) = 0.10x + 6

grams; Total salt =

0.22x + 6

grams.

To find the total amount of salt before mixing.

3

Express the total mass of the mixture and the total salt content using the final mixture's percentage.

Total mass of mixture =

\frac{4}{3}x + 20

grams; Total salt in final mixture =

0.18\left(\frac{4}{3}x + 20\right) = 0.24x + 3.6

grams.

To write the total salt content in terms of the final mixture's concentration.

4

Equate the two expressions for the total mass of salt and solve the linear equation for

x

.

0.22x + 6 = 0.24x + 3.6 \implies 2.4 = 0.02x \implies x = 120

.

To find the mass of Solution A that satisfies the mixture conditions.

Key Concept

Setting up and solving a linear equation in one variable from a mixture word problem.

Question 97Question →

For the linear function $f$ , the table below displays selected values of $x$ and their corresponding function values $f(x)$ .

$x$	$f(x)$
$-2$	$1$
$2$	$9$
$4$	$13$

What is the value of $f(10)$ ?

Show answer & explanation

Answer: 25

Answer

The value of

f(10)

is 25.

The correct answer is 25. By using the points

(-2, 1)

and

(2, 9)

, the slope of the linear function is calculated as

m = \frac{9 - 1}{2 - (-2)} = 2

. Substituting this slope and the point

(2, 9)

into the slope-intercept form,

f(x) = 2x + b

, gives

9 = 2(2) + b

, which resolves to

b = 5

. Thus, the linear function is defined by

f(x) = 2x + 5

. Evaluating the function at

x = 10

yields

f(10) = 2(10) + 5 = 25

.

Step-by-Step Solution

1

Calculate the slope of the linear function using two coordinate pairs.

Slope

m = 2

The slope of a linear function can be determined by the formula

m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

. Substituting

(-2, 1)

and

(2, 9)

yields

m = \frac{9 - 1}{2 - (-2)} = \frac{8}{4} = 2

.

2

Find the y-intercept of the function to write its equation.

f(x) = 2x + 5

Substituting the slope

m = 2

and the point

(2, 9)

into the slope-intercept form

f(x) = mx + b

gives

9 = 2(2) + b

, which simplifies to

b = 5

.

3

Evaluate the function for the input value 10.

f(10) = 25

Substituting

x = 10

into the linear function equation

f(x) = 2x + 5

yields

f(10) = 2(10) + 5 = 25

.

Key Concept

Determining a linear function from a table of values and using it to find specific outputs.

Estimated Time:1m 30s

Question 98Question →

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

Show answer & explanation

Answer: 36

Answer

The correct answer is 36.

To solve the equation

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

, we first clear the fractions by multiplying the entire equation by the least common denominator, 6, which yields

5(y - 2) - 2(2y - 5) = 9

. Distributing the terms on the left side gives

5y - 10 - 4y + 10 = 9

. Combining like terms simplifies this to

y = 9

. The question asks for the value of

4y

, so we multiply

9

by

4

to get the final answer of

36

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator, which is 6.

5(y - 2) - 2(2y - 5) = 9

Multiplying by 6 eliminates the fractions, simplifying the equation.

2

Distribute the coefficients across the parentheses.

5y - 10 - 4y + 10 = 9

Expanding the terms allows like terms to be combined in the next step.

3

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

y = 9

Combining

5y - 4y

yields

y

, and combining

-10 + 10

yields

0

, isolating the variable.

4

Multiply the value of

y

by 4.

4y = 36

The question asks for the value of

4y

rather than just

y

.

Key Concept

Solving a linear equation in one variable by clearing fractions and combining like terms.

Question 99Question →

A line in the $xy$ -plane is defined by the equation $5y - 2x = 15$ . What is the slope of the line?

\frac{5}{2}

-\frac{2}{5}

\frac{2}{5}

3

Show answer & explanation

Answer: $\frac{2}{5}$

Answer

The slope of the line is

\frac{2}{5}

.

To find the slope of the line, convert the equation

5y - 2x = 15

into slope-intercept form,

y = mx + b

, where

m

represents the slope. First, add

2x

to both sides to get

5y = 2x + 15

. Next, divide both sides by

5

to isolate

y

, yielding

y = \frac{2}{5}x + 3

. In this form, the coefficient of

x

is the slope, which is

\frac{2}{5}

.

Step-by-Step Solution

1

Write the given equation of the line.

5y - 2x = 15

To identify the slope, we need to manipulate the given equation.

2

Isolate the

y

term on one side by adding

2x

to both sides.

5y = 2x + 15

This is the first step in converting the equation to slope-intercept form (

y = mx + b

).

3

Divide both sides of the equation by

5

to solve for

y

.

y = \frac{2}{5}x + 3

Dividing isolates

y

completely, putting the equation in the form

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

4

Identify the slope

m

from the equation.

The slope

m

is

\frac{2}{5}

.

In the equation

y = \frac{2}{5}x + 3

, the coefficient of

x

represents the slope of the line.

Key Concept

Slope of a line from its equation

Question 100Question →

A linear equation models the total cost, in dollars, of purchasing $x$ pounds of almonds and $y$ pounds of walnuts. Under the original pricing, purchasing $8$ pounds of almonds and $15$ pounds of walnuts costs $c$ dollars. If the price per pound of almonds is increased by $25\%$ and the price per pound of walnuts is decreased by $10\%$ , the cost of purchasing $8$ pounds of almonds and $15$ pounds of walnuts is still $c$ dollars. Under the original pricing, a customer can purchase exactly $22$ pounds of almonds and no walnuts for $c$ dollars. How many pounds of walnuts and no almonds can the customer purchase for $c$ dollars under the original pricing?

Show answer & explanation

Answer: 16.5

Answer

16.5

Under the original pricing, the cost of

8

pounds of almonds at

a

dollars per pound and

15

pounds of walnuts at

b

dollars per pound is represented by the linear equation

8a + 15b = c

. Under promotional pricing, the cost of almonds increases by

25\%

to

1.25a

, and the cost of walnuts decreases by

10\%

to

0.90b

. The new cost for the same amounts of nuts is

8(1.25a) + 15(0.90b) = 10a + 13.5b = c

. Setting the two cost equations equal to each other gives

8a + 15b = 10a + 13.5b

, which simplifies to

1.5b = 2a

, or

a = 0.75b

. Since

22

pounds of almonds can be purchased for

c

dollars, we have

22a = c

. Substituting

a = 0.75b

yields

22(0.75b) = c

, which simplifies to

16.5b = c

. Thus, exactly

16.5

pounds of walnuts can be purchased for

c

dollars.

Step-by-Step Solution

1

Set up the equation representing the total cost of the initial purchase under the original pricing.

8a + 15b = c

, where

a

is the original price per pound of almonds and

b

is the original price per pound of walnuts.

To represent the cost relation using the original individual prices of almonds and walnuts.

2

Modify the individual prices for the promotional rates and write the new total cost equation.

8(1.25a) + 15(0.90b) = c \implies 10a + 13.5b = c

.

The price per pound of almonds increases by

25\%

(multiplied by

1.25

) and the price per pound of walnuts decreases by

10\%

(multiplied by

0.90

).

3

Equate the two expressions representing

c

to find the ratio between the prices

a

and

b

.

8a + 15b = 10a + 13.5b \implies 2a = 1.5b \implies a = 0.75b

.

Since both purchasing combinations yield the same total budget

c

, their cost equations are equal.

4

Use the budget equation for purchasing only almonds to determine the equivalent purchase in walnuts.

22a = c \implies 22(0.75b) = c \implies 16.5b = c

. Therefore,

16.5

pounds of walnuts can be bought for

c

dollars.

Substituting the price relationship

a = 0.75b

allows expressing the budget

c

purely in terms of the price of walnuts

b

.

Key Concept

Modeling linear relationships in two variables and analyzing changes in coefficients.

Algebra

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