Algebra Practice Questions — SAT — Page 3

In a science experiment, the temperature

y

, in degrees Celsius, and the time

x

, in minutes, are related by a system of equations. The relationships are given by:

y = 2x - 3

3x - y = 8

What is the value of

x

in the solution to this system?

1

5

7

11

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. Substituting the expression for

y

from the first equation into the second equation yields

3x - (2x - 3) = 8

. Distributing the negative sign results in

3x - 2x + 3 = 8

. Combining the variable terms gives

x + 3 = 8

. Subtracting 3 from both sides isolates

x

, giving

x = 5

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation,

y = 2x - 3

, into the second equation.

3x - (2x - 3) = 8

This eliminates the variable

y

so the equation can be solved directly for

x

.

2

Distribute the negative sign to both terms inside the parentheses.

3x - 2x + 3 = 8

Distributing the negative sign to both

2x

and

-3

simplifies the expression correctly.

3

Combine like terms on the left side of the equation.

x + 3 = 8

Subtracting

2x

from

3x

leaves a single

x

term.

4

Subtract 3 from both sides of the equation to isolate

x

.

x = 5

This isolates the variable

x

to find the final solution.

Key Concept

Solving systems of linear equations using substitution

Estimated Time:1m 0s

Question 42Question →

In the equation below,

a

and

b

are constants.

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

If the equation has infinitely many solutions for

x

, what is the value of

a - b

?

9

10

15

17

Show answer & explanation

Answer: 15

Answer

The value of

a - b

is 15.

The correct answer is 15. Multiplying the entire equation by 6 to clear the denominators, distributing the terms, and grouping them yields

(2a - 12)x - (2a + 4b) = (a + 4)x - (ab + 20)

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

x

on both sides must be equal (

2a - 12 = a + 4

), and the constant terms must also be equal (

2a + 4b = ab + 20

). Solving this system of equations yields

a = 16

and

b = 1

. The difference

a - b

is

16 - 1 = 15

.

Step-by-Step Solution

1

Eliminate the denominators by multiplying both sides of the equation by 6.

2[a(x - 1) - 2(3x + b)] = 4(x - 5) - a(b - x)

Multiplying both sides by the least common multiple of 3 and 6 simplifies the equation by removing the fractions.

2

Expand both sides of the equation using the distributive property.

2[ax - a - 6x - 2b] = 4x - 20 - ab + ax

which simplifies to

2ax - 2a - 12x - 4b = ax + 4x - ab - 20

Distributing the constants outside the parentheses allows us to group like terms.

3

Group the

x

-terms and constant terms on each side of the equation.

(2a - 12)x - (2a + 4b) = (a + 4)x - (ab + 20)

Structuring the equation in the form

Ax + B = Cx + D

makes it easier to compare the coefficients.

4

Set up a system of equations by equating the coefficients of

x

and the constant terms from both sides.

2a - 12 = a + 4

and

2a + 4b = ab + 20

For a linear equation to have infinitely many solutions, the coefficient of

x

on both sides must be equal, and the constant terms on both sides must also be equal.

5

Solve the first equation for

a

.

a = 16

Subtracting

a

and adding 12 to both sides isolates the variable

a

.

6

Substitute

a = 16

into the second equation and solve for

b

.

2(16) + 4b = 16b + 20 \implies 32 + 4b = 16b + 20 \implies 12 = 12b \implies b = 1

Substituting the known value of

a

leaves a single linear equation in terms of

b

.

7

Calculate the value of

a - b

.

16 - 1 = 15

This is the final value requested by the question.

Key Concept

Solving linear equations in one variable with infinitely many solutions by equating coefficients on both sides of the equation.

Question 43Question →

A water tank contains 24 gallons of water. Water is being drained from the tank at a constant rate. After 8 minutes, the tank contains 14 gallons of water. If the volume of water in the tank, in gallons, is a linear function of the time in minutes, how many minutes will it take for the tank to contain exactly 4 gallons of water?

Show answer & explanation

Answer: 16

Answer

16

The volume of water in the tank decreases linearly from an initial value of

24

gallons at

t = 0

to

14

gallons at

t = 8

. The constant rate of change (slope) is calculated by dividing the change in volume by the change in time:

\frac{14 - 24}{8 - 0} = -1.25

gallons per minute. Using the slope-intercept form, the volume

V

at time

t

is given by

V = -1.25t + 24

. Setting the volume

V = 4

gives the equation

4 = -1.25t + 24

. Solving for

t

yields

-20 = -1.25t

, which simplifies to

t = 16

.

Step-by-Step Solution

1

Determine the initial state and the state after 8 minutes as coordinate points.

The initial state is

(0, 24)

and the state after

8

minutes is

(8, 14)

.

These coordinates represent the relationship between time and volume of water in the tank.

2

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

Slope

m = \frac{14 - 24}{8 - 0} = -1.25

The slope represents the constant rate at which water is being drained from the tank.

3

Formulate the linear equation.

V(t) = -1.25t + 24

Using the slope-intercept form

V(t) = mt + b

, where

b = 24

is the vertical intercept representing the initial volume.

4

Solve for the time

t

when the volume of water is

4

gallons.

4 = -1.25t + 24 \implies -20 = -1.25t \implies t = 16

To find the time at which the volume decreases to exactly

4

gallons.

Key Concept

Linear Functions and Graphs

Question 44Question →

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

2

4

8

-2

Show answer & explanation

Answer: 8

Answer

The value of

x

is

8

.

To solve the equation, distribute the 5 to the terms inside the parentheses to get

5x - 15 - 2x = 9

. Combine the like terms

5x

and

-2x

to simplify the equation to

3x - 15 = 9

. Add 15 to both sides to isolate the variable term, which yields

3x = 24

. Finally, divide both sides by 3 to find

x = 8

.

Step-by-Step Solution

1

Distribute the

5

to the terms inside the parentheses.

5x - 15 - 2x = 9

To remove the parentheses and begin simplifying the equation.

2

Combine the like variable terms

5x

and

-2x

.

3x - 15 = 9

To group the x terms together.

3

Add

15

to both sides of the equation.

3x = 24

To isolate the term with the variable on one side.

4

Divide both sides of the equation by

3

.

x = 8

To solve for

x

.

Key Concept

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

Estimated Time:45s

Question 45Question →

If $\frac{2}{5}(3x - 4) - \frac{1}{3}(2x + 5) = \frac{1}{5}x + \frac{11}{15}$ , what is the value of $2x - 7$ ?

Show answer & explanation

Answer: 17

Answer

The correct answer is 17.

To find the value of

2x - 7

, first solve the linear equation for

x

. Distributing the coefficients on the left side of the equation gives

\frac{6}{5}x - \frac{8}{5} - \frac{2}{3}x - \frac{5}{3} = \frac{1}{5}x + \frac{11}{15}

. Combining the variable terms and constants on the left side results in

\frac{8}{15}x - \frac{49}{15} = \frac{3}{15}x + \frac{11}{15}

. Subtracting

\frac{3}{15}x

and adding

\frac{49}{15}

to both sides yields

\frac{5}{15}x = \frac{60}{15}

, which simplifies to

\frac{1}{3}x = 4

, or

x = 12

. Finally, substituting 12 into the expression

2x - 7

gives

2(12) - 7 = 17

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses.

\frac{6}{5}x - \frac{8}{5} - \frac{2}{3}x - \frac{5}{3} = \frac{1}{5}x + \frac{11}{15}

To eliminate parentheses and allow grouping of like terms.

2

Combine the variable terms and the constant terms on the left side using a common denominator of 15.

\frac{8}{15}x - \frac{49}{15} = \frac{3}{15}x + \frac{11}{15}

To simplify the linear equation into a standard two-sided form.

3

Subtract the variable term from the right side and add the constant term from the left side.

\frac{5}{15}x = \frac{60}{15}

, which simplifies to

x = 12

To isolate the variable

x

on one side of the equation.

4

Evaluate the expression

2x - 7

using the value of

x

.

2(12) - 7 = 17

To solve for the final requested quantity.

Key Concept

Linear Equations in One Variable

Question 46Question →

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

$x$	$f(x)$
$2$	$11$
$4$	$17$
$6$	$23$

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

Show answer & explanation

Answer: 5

Answer

The correct answer is 5, representing the y-coordinate of the y-intercept of the graph of f.

To find the

y

-intercept of the linear function, we first determine its slope using the points

(2, 11)

and

(4, 17)

from the table. The slope

m

is

\frac{17 - 11}{4 - 2} = \frac{6}{2} = 3

. Next, we use the slope-intercept equation

f(x) = mx + b

. Substituting

m = 3

and the point

(2, 11)

gives

11 = 3(2) + b

, which simplifies to

11 = 6 + b

. Solving for

b

yields

5

. Therefore, the

y

-coordinate of the

y

-intercept of the graph of

f

is

5

.

Step-by-Step Solution

1

Calculate the slope (

m

) of the linear function using two coordinate pairs from the table.

The slope is

m = 3

.

A linear function has a constant slope, which can be found using the formula

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

.

2

Substitute the slope and one of the points into the slope-intercept equation

f(x) = mx + b

to solve for the

y

-intercept

b

.

The

y

-intercept

b

is

5

.

The

y

-coordinate of the

y

-intercept of the graph of

y = f(x)

is the value of

b

in the equation

f(x) = mx + b

.

Key Concept

Finding the y-intercept of a linear function from a table of values.

Estimated Time:45s

Question 47Question →

In the $xy$ -plane, line $l_1$ passes through the origin and has a positive slope. Line $l_2$ is perpendicular to $l_1$ and intersects the $x$ -axis at $(d, 0)$ , where $d > 0$ . The two lines intersect at the point $(a, b)$ in the first quadrant. If the ratio of $a$ to $b$ is $3$ to $4$ , and the area of the triangle bounded by $l_1$ , $l_2$ , and the $x$ -axis is $A$ , which of the following expressions represents $d$ in terms of $A$ ?

\frac{5\sqrt{6A}}{8}

\frac{25A}{6}

\frac{5\sqrt{6A}}{6}

\frac{10\sqrt{6A}}{9}

Show answer & explanation

Answer: $\frac{5\sqrt{6A}}{6}$

Answer

The expression

\frac{5\sqrt{6A}}{6}

represents

d

in terms of

A

.

The correct expression is

\frac{5\sqrt{6A}}{6}

. The slope of line

l_1

passing through the origin and

(a,b)

is given by

m_1 = \frac{b}{a}

. Since the ratio of

a

to

b

is

3

to

4

, we have

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

, which implies the slope

m_1 = \frac{4}{3}

. Because line

l_2

is perpendicular to

l_1

, its slope is

m_2 = -\frac{3}{4}

. The equation of

l_2

passing through

(d,0)

is

y = -\frac{3}{4}(x - d)

. Solving the system of equations gives the intersection coordinates

a = \frac{9}{25}d

and

b = \frac{12}{25}d

. The area of the triangle bounded by the two lines and the

x

-axis is

A = \frac{1}{2} \times d \times b = \frac{6}{25}d^2

. Solving for

d

yields

d = \sqrt{\frac{25A}{6}} = \frac{5\sqrt{6A}}{6}

.

Step-by-Step Solution

1

Determine the slope and equation of line

l_1

.

Slope

m_1 = \frac{4}{3}

, and the equation of the line is

y = \frac{4}{3}x

.

Since

l_1

passes through

(0,0)

and the point

(a,b)

in the first quadrant, its slope is

m_1 = \frac{b}{a}

. Since the ratio of

a

to

b

is

3

to

4

, we have

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

.

2

Determine the equation of line

l_2

.

The equation of the line is

y = -\frac{3}{4}(x - d)

.

Line

l_2

is perpendicular to

l_1

, so its slope is the negative reciprocal of

m_1

, which is

m_2 = -\frac{3}{4}

. Using the point-slope form with the

x

-intercept

(d,0)

, the equation is

y - 0 = -\frac{3}{4}(x - d)

.

3

Find the coordinates of the intersection point

(a,b)

in terms of

d

.

a = \frac{9}{25}d

and

b = \frac{12}{25}d

.

Set the two equations equal to find the

x

-coordinate of the intersection:

\frac{4}{3}x = -\frac{3}{4}(x - d) \implies \frac{4}{3}x = -\frac{3}{4}x + \frac{3}{4}d \implies \frac{25}{12}x = \frac{3}{4}d \implies x = \frac{9}{25}d

. Substituting this back into the equation of

l_1

yields the

y

-coordinate:

y = \frac{4}{3}\left(\frac{9}{25}d\right) = \frac{12}{25}d

.

4

Express the area of the triangle in terms of

d

.

A = \frac{6}{25}d^2

.

The base of the triangle along the

x

-axis is

d

(from

x=0

to

x=d

). The height of the triangle is the

y

-coordinate of the intersection point,

b = \frac{12}{25}d

. The area of the triangle is given by

A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}d\left(\frac{12}{25}d\right) = \frac{6}{25}d^2

.

5

Solve for

d

in terms of

A

.

d = \frac{5\sqrt{6A}}{6}

.

Isolate

d^2

to get

d^2 = \frac{25A}{6}

. Taking the square root of both sides gives

d = \sqrt{\frac{25A}{6}} = \frac{5\sqrt{A}}{\sqrt{6}} = \frac{5\sqrt{6A}}{6}

.

Key Concept

Using the properties of perpendicular lines, setting up equations from coordinate parameters, and applying geometric formulas to relate parameters in linear systems.

Alternative Method

Using the geometric mean theorem (altitude rule) in a right triangle, the altitude

b

divides the hypotenuse

d

into segments

a

and

d-a

, such that

b^2 = a(d-a)

. Since

b = \frac{4}{3}a

, we can substitute this to find

a = \frac{9}{25}d

and

b = \frac{12}{25}d

directly without finding the line equations. Then,

A = \frac{1}{2} d b = \frac{6}{25}d^2

, which solves to

d = \frac{5\sqrt{6A}}{6}

.

Estimated Time:3m 0s

Question 48Question →

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $3x + 4y = 48$ . The graph of $f$ intersects the $x$ -axis at the point $(p, 0)$ and the $y$ -axis at the point $(0, q)$ , where $q > 0$ . If the distance between the points $(p, 0)$ and $(0, q)$ is $15$ , what is the value of $q$ ?

Show answer & explanation

Answer: 12

Answer

12

First, find the slope of the given line by rewriting

3x + 4y = 48

in slope-intercept form:

y = -\frac{3}{4}x + 12

. The slope is

-\frac{3}{4}

. The graph of the linear function

f

is perpendicular to this line, so its slope is the negative reciprocal,

\frac{4}{3}

. With a

y

-intercept of

(0, q)

, the equation of

f

is

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

. Setting

y = 0

gives the

x

-intercept

(p, 0) = (-\frac{3}{4}q, 0)

. The distance between these intercepts is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

(since

q > 0

). Given that the distance is

15

, we solve

\frac{5}{4}q = 15

to find

q = 12

.

Step-by-Step Solution

1

Find the slope of the line

3x + 4y = 48

.

The slope of the line is

-\frac{3}{4}

.

To find the slope of the perpendicular line

f

, we first need the slope of the given line. Rewriting

3x + 4y = 48

in slope-intercept form gives

y = -\frac{3}{4}x + 12

.

2

Determine the slope of

f

.

The slope of

f

is

\frac{4}{3}

.

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

3

Find the

x

-intercept of

f

in terms of

q

.

The

x

-intercept of

f

is

(-\frac{3}{4}q, 0)

.

Since the

y

-intercept of

f

is

(0, q)

, the equation of

f

is

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

. Setting

y = 0

gives

0 = \frac{4}{3}x + q

, which simplifies to

x = -\frac{3}{4}q

.

4

Use the distance formula between the intercepts to solve for

q

.

The value of

q

is

12

.

The distance between

(-\frac{3}{4}q, 0)

and

(0, q)

is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

since

q > 0

. Setting this distance to

15

gives

\frac{5}{4}q = 15

, which yields

q = 12

.

Key Concept

Properties of perpendicular lines, finding intercepts, and utilizing the distance formula in coordinate geometry.

Question 49Question →

An online store charges a flat shipping fee of $8.50$ dollars for orders under $100$ dollars. A customer purchases $5$ identical shirts and uses a coupon for $12.00$ dollars off the total price of the shirts. If the total charge for the order, including shipping, is $86.50$ dollars, what is the price, in dollars, of one shirt?

13.20

15.60

18.00

19.70

Show answer & explanation

Answer: $18.00$

Answer

The price of one shirt is

18.00

dollars.

To find the price of one shirt, we set up a linear equation. Let

s

represent the price of one shirt. The cost of

5

shirts after applying the

12.00

dollar coupon is

5s - 12.00

. Since this amount is under

100

dollars, the flat shipping fee of

8.50

dollars is added to the total. This gives the equation

5s - 12.00 + 8.50 = 86.50

. Combining the constants on the left side results in

5s - 3.50 = 86.50

. Adding

3.50

to both sides yields

5s = 90.00

. Dividing both sides by

5

gives

s = 18.00

. Thus, the price of one shirt is

18.00

dollars.

Step-by-Step Solution

1

Represent the cost of the shirts using a variable.

Let

s

represent the price of one shirt. The cost of

5

shirts with the

12.00

dollar coupon discount is

5s - 12.00

dollars.

This establishes the variable for the unknown value we need to find.

2

Set up the full linear equation by including the shipping fee and total charge.

Since

5s - 12.00

is under

100

dollars (which we verify at the end), a flat shipping fee of

8.50

dollars applies. The equation is

5s - 12.00 + 8.50 = 86.50

.

This relates all parts of the word problem into a single solvable mathematical statement.

3

Simplify the equation by combining the constant values on the left side.

5s - 3.50 = 86.50

Combining like terms simplifies the algebraic expression.

4

Isolate the variable term by adding

3.50

to both sides.

5s = 90.00

This moves all constant terms to one side of the equation.

5

Solve for the variable by dividing both sides of the equation by

5

.

s = 18.00

Dividing by the coefficient of the variable isolates

s

to find the final value.

Key Concept

Linear Equations in One Variable

Question 50Question →

In the $xy$ -plane, the graph of a linear function $f$ has a $y$ -intercept of $(0, r)$ and an $x$ -intercept of $(s, 0)$ , where $r$ and $s$ are nonzero constants. A second line, $g$ , is perpendicular to the graph of $f$ and passes through the point $(s, r)$ . Which of the following expressions represents the $x$ -intercept of line $g$ ?

\frac{s^2 - r^2}{r}

\frac{s^2 + r^2}{s}

\frac{s^2 - r^2}{s}

\frac{r^2 - s^2}{r}

Show answer & explanation

Answer: $\frac{s^2 - r^2}{s}$

Answer

The expression

\frac{s^2 - r^2}{s}

The slope of line

f

is found using its intercepts

(0, r)

and

(s, 0)

, which gives a slope of

m_f = -\frac{r}{s}

. Since line

g

is perpendicular to line

f

, its slope is the negative reciprocal,

m_g = \frac{s}{r}

. Using the point-slope form with the point

(s, r)

, the equation for line

g

is

y - r = \frac{s}{r}(x - s)

. Setting

y = 0

to find the

x

-intercept yields

x = \frac{s^2 - r^2}{s}

.

Step-by-Step Solution

1

Determine the slope of the linear function

f

using its intercepts.

The slope of

f

is

m_f = \frac{0 - r}{s - 0} = -\frac{r}{s}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Find the slope of line

g

, which is perpendicular to the graph of

f

.

The slope of line

g

is

m_g = -\frac{1}{m_f} = -\frac{1}{-r/s} = \frac{s}{r}

.

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

3

Write the equation of line

g

in point-slope form using the point

(s, r)

.

The equation of line

g

is

y - r = \frac{s}{r}(x - s)

.

The point-slope form of a linear equation is

y - y_1 = m(x - x_1)

.

4

Solve for the

x

-intercept of line

g

by setting

y = 0

.

Setting

y = 0

gives

-r = \frac{s}{r}(x - s) \implies -\frac{r^2}{s} = x - s \implies x = s - \frac{r^2}{s} = \frac{s^2 - r^2}{s}

.

The

x

-intercept of a line is the value of

x

when

y = 0

.

Key Concept

Finding the equation of a perpendicular line and its intercepts using algebraic representations.

Question 51Question →

If $3(2n - 5) = 4n + 9$ , what is the value of $n$ ?

Show answer & explanation

Answer: 12

Answer

12

To solve the equation

3(2n - 5) = 4n + 9

, we first distribute the 3 to the terms inside the parentheses to get

6n - 15 = 4n + 9

. Next, we subtract

4n

from both sides to group the variable terms, giving

2n - 15 = 9

. We then add 15 to both sides to isolate the variable term, resulting in

2n = 24

. Finally, dividing both sides by 2 gives the solution

n = 12

.

Step-by-Step Solution

1

Distribute 3 to the terms inside the parentheses.

6n - 15 = 4n + 9

To simplify the left side of the equation and remove parentheses.

2

Subtract

4n

from both sides of the equation.

2n - 15 = 9

To group the variable terms on one side of the equation.

3

Add 15 to both sides of the equation.

2n = 24

To isolate the term with the variable

n

.

4

Divide both sides of the equation by 2.

n = 12

To solve for

n

.

Key Concept

Solving a linear equation in one variable by applying the distributive property and isolating the variable.

Question 52Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(-2, c)$ and $(c, 7)$ , where $c$ is a constant. If the slope of the graph of $f$ is $\frac{1}{2}$ , what is the value of $c$ ?

1

4

6

16

Show answer & explanation

Answer: 4

Answer

The value of

c

is 4.

Using the slope formula

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

with the points

(-2, c)

and

(c, 7)

and the slope

m = \frac{1}{2}

gives

\frac{7 - c}{c - (-2)} = \frac{1}{2}

. Simplifying the denominator yields

\frac{7 - c}{c + 2} = \frac{1}{2}

. Cross-multiplying gives

2(7 - c) = c + 2

, which expands to

14 - 2c = c + 2

. Adding

2c

to both sides and subtracting

2

from both sides gives

12 = 3c

. Dividing both sides by

3

gives

c = 4

.

Step-by-Step Solution

1

Set up the slope formula for the line passing through

(-2, c)

and

(c, 7)

.

m = \frac{7 - c}{c - (-2)}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Substitute the given slope

m = \frac{1}{2}

into the formula and simplify the denominator.

\frac{7 - c}{c + 2} = \frac{1}{2}

We are given that the slope of the line is

\frac{1}{2}

.

3

Solve the equation for

c

by cross-multiplying and isolating the variable.

2(7 - c) = c + 2 \implies 14 - 2c = c + 2 \implies 12 = 3c \implies c = 4

Cross-multiplying eliminates the fractions, allowing us to solve the linear equation for

c

.

Key Concept

Finding the slope of a line given two points containing an unknown coordinate

Question 53Question →

Consider the system of equations below.

x = 2y

3x - y = 10

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

Show answer & explanation

Answer: 4

Answer

The value of

x

is

4

.

The system of equations can be solved by substituting

x = 2y

from the first equation into the second equation, which gives

3(2y) - y = 10

. Simplifying this equation yields

6y - y = 10

, or

5y = 10

. Dividing by

5

gives

y = 2

. Substituting

y = 2

back into the first equation yields

x = 2(2) = 4

. Therefore, the value of

x

is

4

.

Step-by-Step Solution

1

Substitute the expression for

x

from the first equation into the second equation.

3(2y) - y = 10

Since the first equation gives

x

in terms of

y

, substituting it into the second equation reduces the system to a single linear equation in one variable.

2

Solve the resulting equation for

y

.

y = 2

Simplifying

3(2y) - y = 10

yields

6y - y = 10

, which simplifies further to

5y = 10

. Dividing both sides by

5

gives

y = 2

.

3

Substitute the value of

y

back into the first equation to solve for

x

.

x = 4

Using

x = 2y

and substituting

y = 2

gives

x = 2(2) = 4

.

Key Concept

Solving systems of linear equations using the substitution method.

Question 54Question →

In the equation below,

a

and

b

are positive constants.

\frac{2}{3} \left( \frac{3}{2}x - a \right) - \frac{3}{4} \left( 8 - bx \right) = \frac{5}{2}(x - 3) - \frac{1}{2}

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

a^b

?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

The correct answer is 9. Expanding the left side of the equation yields

\left(1 + \frac{3}{4}b\right)x - \left(6 + \frac{2}{3}a\right)

, and simplifying the right side yields

\frac{5}{2}x - 8

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

x

on the left,

1 + \frac{3}{4}b

, must equal the coefficient of

x

on the right,

\frac{5}{2}

, which gives

b = 2

. Similarly, the constant term on the left,

-\left(6 + \frac{2}{3}a\right)

, must equal the constant term on the right,

-8

, which simplifies to

6 + \frac{2}{3}a = 8

and gives

a = 3

. Evaluating

a^b

with these values yields

3^2 = 9

.

Step-by-Step Solution

1

Expand both sides of the equation to collect like terms.

\left( 1 + \frac{3}{4}b \right)x - \left( 6 + \frac{2}{3}a \right) = \frac{5}{2}x - 8

Expanding allows us to compare the coefficient of

x

and the constant term on each side of the equation.

2

Equate the coefficients of

x

on both sides of the equation.

1 + \frac{3}{4}b = \frac{5}{2} \implies b = 2

For a linear equation to have infinitely many solutions, the coefficient of

x

must be identical on both sides.

3

Equate the constant terms on both sides of the equation.

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

For a linear equation to have infinitely many solutions, the constant terms must also be identical on both sides.

4

Calculate the value of

a^b

using the solved values of

a

and

b

.

3^2 = 9

The question asks for the value of the expression

a^b

where

a = 3

and

b = 2

.

Key Concept

A linear equation in one variable of the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

.

Alternative Method

Since the equation must hold for all values of

x

if it has infinitely many solutions, you can substitute convenient values for

x

to solve for

a

and

b

directly. Substituting

x = 0

simplifies the equation to

-\frac{2}{3}a - 6 = -8

, which quickly yields

a = 3

. Then, substituting

x = 2

and

a = 3

simplifies the equation to

-\frac{3}{4}(8 - 2b) = -3

, which yields

b = 2

. Calculating

a^b

gives

3^2 = 9

.

Estimated Time:3m 0s

Question 55Question →

If $\frac{5}{3}(3x - 6) - \frac{1}{2}(4x + 8) = 2$ , what is the value of $3x - 5$ ?

Show answer & explanation

Answer: 11

Answer

11

Distributing the fraction

\frac{5}{3}

to

(3x - 6)

yields

5x - 10

. Distributing the term

-\frac{1}{2}

to

(4x + 8)

yields

-2x - 4

. Combining these results gives

(5x - 2x) + (-10 - 4) = 2

, which simplifies to

3x - 14 = 2

. Adding 14 to both sides yields

3x = 16

. Finally, subtracting 5 from both sides of this equation gives

3x - 5 = 11

.

Step-by-Step Solution

1

Distribute

\frac{5}{3}

and

-\frac{1}{2}

to the terms within their respective parentheses.

5x - 10 - 2x - 4 = 2

This removes the parentheses and allows us to group terms.

2

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

3x - 14 = 2

Simplifying the expression on the left side makes it easier to solve.

3

Add 14 to both sides of the equation to isolate the term

3x

.

3x = 16

Since the question asks for the value of

3x - 5

, isolating

3x

allows direct evaluation.

4

Subtract 5 from both sides of the equation

3x = 16

.

3x - 5 = 11

This directly yields the required value without needing to compute the fractional value of

x

first.

Key Concept

Solving linear equations in one variable by distribution, combining like terms, and evaluating expressions.

Estimated Time:1m 30s

Question 56Question →

In the $xy$ -plane, the graph of a linear function $f$ has a positive $y$ -intercept and a positive $x$ -intercept. The area of the triangular region in the first quadrant bounded by the graph of $f$ and the coordinate axes is $36$ . If the graph of $f$ passes through the point $(4, 4)$ and has a slope less than $-1$ , what is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 12

Answer

The

y

-coordinate of the

y

-intercept of the graph of

f

is

12

.

The correct answer is

12

because the system of equations derived from the area constraint (

qr = 72

) and the point constraint (

q+r=18

) yields two possible values for the

y

-intercept:

6

or

12

. The condition that the slope must be less than

-1

means that the line must be steeper than a slope of

-1

, which requires the

y

-intercept to be larger than the

x

-intercept (

r > q

). Thus, the

y

-intercept is

12

.

Step-by-Step Solution

1

Express the equation of the line using intercept form.

\frac{x}{q} + \frac{y}{r} = 1

, where

q > 0

is the

x

-intercept and

r > 0

is the

y

-intercept.

Since the line intersects the positive axes, this form directly relates the intercepts to the coordinates of points on the line.

2

Use the area of the triangle to find a relationship between

q

and

r

.

qr = 72

The area of the right triangle formed by the axes and the intercepts is given by

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}qr = 36

.

3

Substitute the point

(4, 4)

into the equation of the line and simplify using the area relationship.

q + r = 18

Plugging in

x = 4

and

y = 4

yields

\frac{4}{q} + \frac{4}{r} = 1 \implies 4(q + r) = qr

. Substituting

qr = 72

gives

4(q + r) = 72 \implies q + r = 18

.

4

Solve the system of equations

q + r = 18

and

qr = 72

.

(q, r) = (6, 12)

or

(q, r) = (12, 6)

Substituting

r = 18 - q

into

qr = 72

yields

q(18 - q) = 72 \implies q^2 - 18q + 72 = 0 \implies (q - 6)(q - 12) = 0

.

5

Apply the slope condition to determine the unique value of

r

.

r = 12

The slope of the line is

m = -\frac{r}{q}

. If

q = 12

and

r = 6

, then

m = -0.5

, which is not less than

-1

. If

q = 6

and

r = 12

, then

m = -2

, which is less than

-1

. Thus,

r = 12

is the correct

y

-intercept.

Key Concept

Using intercepts and area to determine the equation of a linear function under constraints.

Question 57Question →

In the equation $3(2x - a) - 4(x + 1) = 2x + 5$ , $a$ is a constant. If the equation has infinitely many solutions, what is the value of $a$ ?

-\frac{1}{3}

-3

3

-9

Show answer & explanation

Answer: $-3$

Answer

The value of

a

must be

-3

.

The value of

-3

is correct. Expanding and simplifying the left side of the equation yields

2x - 3a - 4 = 2x + 5

. For a linear equation to have infinitely many solutions, the constant terms on both sides must be identical. Setting

-3a - 4 = 5

and solving for

a

gives

-3a = 9

, which simplifies to

a = -3

.

Step-by-Step Solution

1

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

6x - 3a - 4x - 4 = 2x + 5

Distributing eliminates the parentheses, allowing like terms to be combined.

2

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

2x - 3a - 4 = 2x + 5

Grouping the

x

terms (

6x - 4x = 2x

) simplifies the structure of the equation.

3

Set up the condition for the equation to have infinitely many solutions.

-3a - 4 = 5

For a linear equation to have infinitely many solutions, the coefficients of

x

on both sides must be equal, and the constant terms must also be equal. The

x

coefficients are already equal (

2 = 2

), so we equate the constants.

4

Isolate the constant

a

by adding

4

to both sides and then dividing by

-3

.

a = -3

Adding

4

yields

-3a = 9

, and dividing by

-3

isolates the variable to find the solution.

Key Concept

Linear Equations in One Variable

Alternative Method

Instead of fully simplifying the equation, substitute

x = 0

directly into the original equation, since a statement with infinitely many solutions must hold true for all values of

x

. Substituting

x = 0

gives

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

, which simplifies directly to

-3a - 4 = 5

, yielding

a = -3

.

Estimated Time:1m 15s

Question 58Question →

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

$x$	$f(x)$
3	7
6	12
9	17

What is the value of $f(15)$ ?

Show answer & explanation

Answer: 27

Answer

The value of the linear function evaluated at 15 is 27.

To find

f(15)

for the linear function, we determine its constant rate of change. Using points

(3, 7)

and

(6, 12)

, the change in

f(x)

is

12 - 7 = 5

for a change in

x

of

6 - 3 = 3

. This gives a slope of

\frac{5}{3}

. Using point-slope form with

(3, 7)

gives the equation

f(x) - 7 = \frac{5}{3}(x - 3)

, which simplifies to

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

. Substituting

x = 15

yields

f(15) = \frac{5}{3}(15) + 2 = 25 + 2 = 27

. Alternatively, since

x

increases by 6 from 9 to 15, which is twice the step size of 3, the value of

f(x)

must increase by

2 \times 5 = 10

from

17

, resulting in

17 + 10 = 27

.

Step-by-Step Solution

1

Calculate the slope of the linear function

slope

m = \frac{5}{3}

Linear functions have a constant rate of change, which is the slope.

2

Determine the equation of the function

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

Using the slope and one point from the table helps define the function for all inputs.

3

Evaluate the function at

x = 15

f(15) = 27

Substitute the given input value into the function equation to find the corresponding output value.

Key Concept

Linear Functions and Graphs

Question 59Question →

The variables $x$ and $y$ satisfy the system of equations below.

y = 10 - 2x

3x - y = 5

What is the value of $x$ ?

15

4

3

1

Show answer & explanation

Answer: 3

Answer

3

Substituting the expression for

y

from the first equation,

10 - 2x

, into the second equation yields

3x - (10 - 2x) = 5

. Distributing the negative sign results in

3x - 10 + 2x = 5

. Combining the like terms of

x

gives

5x - 10 = 5

. Adding 10 to both sides of the equation results in

5x = 15

. Finally, dividing both sides by 5 gives

x = 3

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

3x - (10 - 2x) = 5

To eliminate the variable

y

and obtain a single linear equation in terms of

x

.

2

Distribute the negative sign through the parentheses and combine like terms.

3x - 10 + 2x = 5

, which simplifies to

5x - 10 = 5

To simplify the linear equation before isolating the variable.

3

Isolate the variable

x

by adding 10 to both sides and then dividing by 5.

5x = 15

, which gives

x = 3

To determine the value of

x

.

Key Concept

Solving systems of linear equations using substitution

Estimated Time:45s

Question 60Question →

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

3x - y = 7

x + y = 5

Show answer & explanation

Answer: 3

Answer

The value of

x

is

3

.

Adding the two equations eliminates

y

, resulting in

4x = 12

. Dividing by

4

gives

x = 3

.

Step-by-Step Solution

1

Add the two equations to eliminate

y

.

4x = 12

Adding the equations

(3x - y) + (x + y) = 7 + 5

eliminates

y

because

-y + y = 0

.

2

Solve for

x

.

x = 3

Divide both sides of the equation

4x = 12

by

4

.

Key Concept

Solving a system of linear equations by elimination.

Algebra

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