All practice questions

Question 21Question →

A pharmaceutical company produces two strengths of a certain medication, Strength P and Strength Q, by mixing an active ingredient with a liquid base. In Strength P, the ratio of the volume of the active ingredient to the volume of the liquid base is $2$ to $5$ . In Strength Q, the ratio of the volume of the active ingredient to the volume of the liquid base is $3$ to $8$ . A technician prepares a batch of Strength P and a batch of Strength Q such that the volume of the liquid base used in the batch of Strength P is equal to the volume of the liquid base used in the batch of Strength Q. If the total volume of the active ingredient used in both batches combined is $62$ liters, what is the total volume of the liquid base, in liters, used in both batches combined?

Show answer & explanation

Answer: 160

Answer

The total volume of the liquid base used in both batches combined is 160 liters.

The total volume of the liquid base used in both batches combined is 160 liters. To find this, let the volume of the liquid base in each batch be

L

liters. The volume of the active ingredient in Strength P is

\frac{2}{5}L

liters, and the volume of the active ingredient in Strength Q is

\frac{3}{8}L

liters. The sum of the active ingredients is

\frac{2}{5}L + \frac{3}{8}L = 62

liters. Finding a common denominator gives

\frac{16}{40}L + \frac{15}{40}L = 62

, which simplifies to

\frac{31}{40}L = 62

. Solving for

L

yields

L = 62 \times \frac{40}{31} = 80

liters. Since both batches contain equal volumes of liquid base, the combined total volume of the liquid base is

2L = 2 \times 80 = 160

liters.

Step-by-Step Solution

1

Represent the volumes of the ingredients in terms of a single variable.

Let the volume of the liquid base used in each batch be

40x

liters, where 40 is chosen as the least common multiple of 5 and 8. The volume of the active ingredient in Strength P is

\frac{2}{5} \times 40x = 16x

liters, and the volume of the active ingredient in Strength Q is

\frac{3}{8} \times 40x = 15x

liters.

This establishes a common algebraic scale for both mixtures, utilizing the fact that the volume of the liquid base is identical in both batches.

2

Formulate an equation using the total volume of the active ingredient.

16x + 15x = 62

, which simplifies to

31x = 62

.

The problem states that the combined volume of the active ingredient from both batches is 62 liters.

3

Solve for the variable

x

.

x = 2

Dividing both sides of the equation by 31 yields the scaling factor.

4

Calculate the total volume of the liquid base used in both batches combined.

Total liquid base =

40x + 40x = 80x = 80(2) = 160

liters.

Since each batch uses

40x

liters of liquid base, the total volume for both batches is the sum of their individual base volumes, which is

80x

liters.

Key Concept

Solving ratio and proportion problems with a shared quantity

Estimated Time:2m 30s

Question 22Question →

If the expression $(x + 4)^2 - (x - 4)^2$ is equivalent to $kx$ for all values of $x$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 16

Answer

The value of the constant

k

is

16

.

To find the value of

k

, the expression

(x + 4)^2 - (x - 4)^2

is simplified by expanding both binomials. The first binomial expands to

x^2 + 8x + 16

, and the second binomial expands to

x^2 - 8x + 16

. Subtracting the second expression from the first requires distributing the negative sign across all terms:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x - (-8x) + 16 - 16 = 16x

. Equating

16x

to

kx

yields

k = 16

.

Step-by-Step Solution

1

Expand the first squared term,

(x + 4)^2

x^2 + 8x + 16

Applying the binomial expansion formula

(a + b)^2 = a^2 + 2ab + b^2

where

a = x

and

b = 4

.

2

Expand the second squared term,

(x - 4)^2

x^2 - 8x + 16

Applying the binomial expansion formula

(a - b)^2 = a^2 - 2ab + b^2

where

a = x

and

b = 4

.

3

Subtract the second expanded expression from the first and simplify

16x

Group like terms and distribute the negative sign:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x + 8x + 16 - 16 = 16x

.

4

Equate the simplified expression to

kx

to determine the constant

k

k = 16

Since

16x = kx

for all values of

x

, the coefficients must be equal.

Key Concept

Equivalent Algebraic Expressions

Question 23Question →

Lines $L_1$ and $L_2$ are parallel. Points $P$ and $Q$ lie on line $L_1$ , and points $R$ and $S$ lie on line $L_2$ . Segment $PS$ and segment $QR$ intersect at point $T$ . If the measure of angle $TPQ$ is $38^\circ$ and the measure of angle $PTQ$ is $105^\circ$ , what is the measure, in degrees, of angle $TRS$ ?

Show answer & explanation

Answer: 37

Answer

37

To find the measure of angle

TRS

, we first find the measure of angle

TQP

using the property that the sum of interior angles in triangle

TPQ

is

180^\circ

. This gives

\angle TQP = 180^\circ - 38^\circ - 105^\circ = 37^\circ

. Since lines

L_1

and

L_2

are parallel, the transversal line

QR

creates alternate interior angles that are equal in measure. Therefore, angle

TRS

is equal to angle

TQP

, which is

37^\circ

.

Step-by-Step Solution

1

Calculate the measure of angle

TQP

in triangle

TPQ

.

\angle TQP = 37^\circ

The sum of the angles in a triangle is

180^\circ

. Therefore,

\angle TQP = 180^\circ - \angle TPQ - \angle PTQ = 180^\circ - 38^\circ - 105^\circ = 37^\circ

.

2

Relate angle

TQP

to angle

TRS

using parallel line properties.

\angle TRS = 37^\circ

Since line

L_1

is parallel to line

L_2

, the transversal line

QR

creates equal alternate interior angles. Thus,

\angle TRS = \angle TQP = 37^\circ

.

Key Concept

Alternate interior angles and triangle angle sum theorem

Question 24Question →

The quadratic function $f$ is defined by $f(x) = -2x^2 + bx + c$ , where $b$ and $c$ are constants. In the $xy$ -plane, the vertex of the graph of $f$ has an $x$ -coordinate of $3$ . If $f(1) = 2$ , what is the $y$ -coordinate of the vertex of the graph of $f$ ?

Show answer & explanation

Answer: 10

Answer

10

The quadratic function in standard form is

f(x) = -2x^2 + bx + c

, which has a leading coefficient of

a = -2

. The vertex form of a quadratic function is

f(x) = a(x-h)^2 + k

, where

(h, k)

is the vertex of the parabola. Given that the

x

-coordinate of the vertex is

3

(so

h = 3

), we can write the function as

f(x) = -2(x-3)^2 + k

. Since the graph passes through the point

(1, 2)

, we substitute

x = 1

and

f(1) = 2

into the equation:

2 = -2(1-3)^2 + k

. Simplifying the expression gives

2 = -2(-2)^2 + k

, which becomes

2 = -2(4) + k

, or

2 = -8 + k

. Adding

8

to both sides yields

k = 10

. Thus, the

y

-coordinate of the vertex is

10

.

Step-by-Step Solution

1

Write the quadratic function in vertex form.

f(x) = -2(x-3)^2 + k

The standard form equation

f(x) = -2x^2 + bx + c

shows that the leading coefficient

a = -2

. The vertex form is

f(x) = a(x-h)^2 + k

, where

(h, k)

is the vertex. Since the

x

-coordinate of the vertex is

3

, we substitute

a = -2

and

h = 3

into the vertex form.

2

Substitute the point

(1, 2)

into the vertex form equation.

2 = -2(1-3)^2 + k

Since

f(1) = 2

, the graph passes through the point

(1, 2)

. Substituting these values allows us to solve for the unknown vertex

y

-coordinate,

k

.

3

Simplify the equation and solve for

k

.

k = 10

Simplifying the term

-2(1-3)^2

yields

-2(-2)^2 = -2(4) = -8

. This simplifies the equation to

2 = -8 + k

. Adding

8

to both sides gives

k = 10

.

Key Concept

Vertex form of a quadratic function

Question 25Question →

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

4x + 3y = 25

2x + 3y = 17

Show answer & explanation

Answer: 3

Answer

The value of

y

is

3

.

Subtracting the second equation from the first equation gives

(4x + 3y) - (2x + 3y) = 25 - 17

, which simplifies to

2x = 8

. Dividing by 2 gives

x = 4

. Substituting

x = 4

into the second equation gives

2(4) + 3y = 17

, which simplifies to

8 + 3y = 17

. Subtracting 8 from both sides gives

3y = 9

, and dividing by 3 gives

y = 3

.

Step-by-Step Solution

1

Subtract the second equation from the first equation to eliminate the

y

term.

2x = 8

Subtracting the equations eliminates

3y

since it is common to both equations, leaving a single variable equation.

2

Solve for

x

by dividing both sides of the equation by 2.

x = 4

Dividing isolates the variable

x

so we can find its numerical value.

3

Substitute

x = 4

into the second equation

2x + 3y = 17

and solve for

y

.

y = 3

Substituting the value of

x

leaves only the variable

y

, which can then be isolated and solved.

Key Concept

Solving systems of linear equations using elimination

Question 26Question →

Two linear equations are defined as follows:

3x + 2y = 12

x - 2y = 4

If the ordered pair $(x, y)$ satisfies both equations, what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Adding the two equations

3x + 2y = 12

and

x - 2y = 4

eliminates the

y

terms, resulting in

4x = 16

. Dividing both sides by 4 gives the value of

x

as 4.

Step-by-Step Solution

1

Add the two equations together to eliminate the

y

variable.

4x = 16

Since the coefficients of

y

are opposites (

2

and

-2

), adding the equations eliminates

y

directly.

2

Divide both sides of the equation by 4.

x = 4

To isolate the variable

x

.

Key Concept

Solving systems of linear equations using the elimination method.

Alternative Method

We can solve the second equation for

x

to get

x = 2y + 4

. Substituting this expression into the first equation gives

3(2y + 4) + 2y = 12

, which simplifies to

6y + 12 + 2y = 12

, or

8y = 0

, meaning

y = 0

. Substituting

y = 0

back into

x = 2y + 4

yields

x = 4

.

Estimated Time:45s

Question 27Question →

In the $xy$ -plane, a right triangle has vertices at the origin $O(0, 0)$ , $A(x, 0)$ , and $B(0, y)$ , where $x > 0$ and $y > 0$ . If the length of the hypotenuse $AB$ is $26$ and the slope of the line containing segment $AB$ is $-\frac{12}{5}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 24

Answer

The value of

y

is 24.

The correct answer is 24 because the slope of the line through

A(x, 0)

and

B(0, y)

is

-\frac{y}{x}

. Setting this equal to

-\frac{12}{5}

gives the relationship

y = \frac{12}{5}x

. Substituting this into the Pythagorean theorem equation

x^2 + y^2 = 26^2

yields

x^2 + \left(\frac{12}{5}x\right)^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

x = 10

, and multiplying by

\frac{12}{5}

gives the value of

y

as 24.

Step-by-Step Solution

1

Express the slope of the line containing segment

AB

in terms of

x

and

y

.

y = \frac{12}{5}x

The line passes through

A(x, 0)

and

B(0, y)

, so its slope is

-\frac{y}{x}

. Setting this equal to the given slope of

-\frac{12}{5}

gives the relationship between

x

and

y

.

2

Apply the Pythagorean theorem to the right triangle

OAB

.

x^2 + y^2 = 676

The lengths of the legs of the right triangle are

x

and

y

, and the hypotenuse is

26

.

3

Substitute the slope relationship into the Pythagorean equation to solve for

x

.

x = 10

Substituting

y = \frac{12}{5}x

gives

x^2 + \frac{144}{25}x^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

10

.

4

Calculate the value of

y

.

y = 24

Using

x = 10

in

y = \frac{12}{5}x

gives

24

.

Key Concept

Solving right triangle problems in the coordinate plane by combining linear equations (slope) with the Pythagorean theorem.

Question 28Question →

A sign is in the shape of a parallelogram with a base of $18$ inches and a height of $10$ inches. A smaller, similar parallelogram is cut out from the center of the sign. If the scale factor of the smaller parallelogram to the larger parallelogram is $1$ to $3$ , what is the area, in square inches, of the remaining portion of the sign?

Show answer & explanation

Answer: 160

Answer

160

The area of the larger parallelogram is

18 \times 10 = 180

square inches. Since the smaller parallelogram is similar to the larger one with a scale factor of

1

to

3

, the area of the smaller parallelogram is

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

of the area of the larger parallelogram. This gives an area of

180 \times \frac{1}{9} = 20

square inches for the smaller parallelogram. Subtracting this from the larger area yields

180 - 20 = 160

square inches.

Step-by-Step Solution

1

Calculate the area of the larger parallelogram.

A_{\text{larger}} = 180

square inches

The area of a parallelogram is the product of its base and height (

A = bh

).

2

Determine the area of the smaller, similar parallelogram.

A_{\text{smaller}} = 20

square inches

The ratio of the areas of similar figures is the square of the scale factor:

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

.

3

Subtract the smaller area from the larger area to find the remaining area.

A_{\text{remaining}} = 160

square inches

The remaining portion of the sign is the total area minus the cutout area.

Key Concept

The area of similar geometric shapes scales by the square of the linear scale factor.

Question 29Question →

A circle with center $O$ has a radius of 15. Chord $AB$ of the circle has a length of 18, and chord $CD$ is parallel to $AB$ and has a length of 24. If the two chords are on opposite sides of the center $O$ , what is the distance between chord $AB$ and chord $CD$ ?

Show answer & explanation

Answer: 21

Answer

21

The distance between the chords is 21. By drawing a line through the center

O

perpendicular to both chords, we bisect chord

AB

at

M

(length 9) and chord

CD

at

N

(length 12). Using the Pythagorean theorem with the radius of 15, the perpendicular distance from the center to chord

AB

is

\sqrt{15^2 - 9^2} = 12

and to chord

CD

is

\sqrt{15^2 - 12^2} = 9

. Since the chords lie on opposite sides of the center, the total distance between them is the sum of these distances,

12 + 9 = 21

.

Step-by-Step Solution

1

Find the perpendicular distance from the center

O

to chord

AB

.

OM = 12

The perpendicular from the center bisects the chord, so we use the Pythagorean theorem in right triangle

OMA

with hypotenuse

OA = 15

and leg

AM = 9

.

2

Find the perpendicular distance from the center

O

to chord

CD

.

ON = 9

Similarly, we use the Pythagorean theorem in right triangle

ONC

with hypotenuse

OC = 15

and leg

CN = 12

.

3

Add the perpendicular distances together to find the total distance between the chords.

21

Because the chords are parallel and on opposite sides of the center, the total distance between them is the sum of their individual distances to the center.

Key Concept

The perpendicular from the center of a circle to a chord bisects the chord, and the distance from the center to the chord can be calculated using the Pythagorean theorem with the radius of the circle.

Question 30Question →

The equation $x^2 + y^2 - 14x - 8y + 40 = 0$ defines a circle in the $xy$ -plane. If the center of the circle has coordinates $(h, k)$ , what is the value of $h + k$ ?

Show answer & explanation

Answer: 11

Answer

The correct answer is 11.

The equation is rewritten in the standard form

(x - 7)^2 + (y - 4)^2 = 25

by completing the square. The center coordinates

(h, k)

are

(7, 4)

. Thus,

h + k = 7 + 4 = 11

.

Step-by-Step Solution

1

Group terms and complete the square for x and y

The equation becomes

(x - 7)^2 + (y - 4)^2 = 25

.

To convert the general form of the circle's equation into the standard form

(x - h)^2 + (y - k)^2 = r^2

.

2

Identify the coordinates of the center

The center

(h, k)

is

(7, 4)

, so

h = 7

and

k = 4

.

The standard form directly gives the center coordinates as

(h, k)

when written as

(x - h)^2 + (y - k)^2 = r^2

.

3

Calculate the sum of the coordinates

h + k = 7 + 4 = 11

The question asks for the value of

h + k

.

Key Concept

Completing the square to find the standard form of a circle's equation and identifying the center coordinates.

Question 31Question →

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

Show answer & explanation

Answer: 7

Answer

7

The correct answer is 7. Expanding the left side of the equation yields

2x - \frac{1}{3} - \frac{3}{2}x + \frac{1}{4} = \frac{1}{2}x - \frac{1}{12}

. Expanding the right side yields

\frac{1}{6}x + \frac{4}{6} - \frac{5}{12} = \frac{1}{6}x + \frac{1}{4}

. Setting the two sides equal gives

\frac{1}{2}x - \frac{1}{12} = \frac{1}{6}x + \frac{1}{4}

. Multiplying all terms by 12 clears the fractions, resulting in

6x - 1 = 2x + 3

. Solving for

x

gives

4x = 4

, which means

x = 1

. Substituting

x = 1

into

12x - 5

yields

12(1) - 5 = 7

.

Step-by-Step Solution

1

Distribute the factors on the left side of the equation:

\frac{2}{3}\left(3x - \frac{1}{2}\right)

and

-\frac{3}{4}\left(2x - \frac{1}{3}\right)

.

2x - \frac{1}{3} - \frac{3}{2}x + \frac{1}{4}

To eliminate the parentheses and prepare the left side of the equation for combining like terms.

2

Combine the variable terms and the constant terms on the left side:

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

.

\frac{1}{2}x - \frac{1}{12}

To simplify the left side into a single linear expression with a common denominator for the constants.

3

Expand and simplify the right side of the equation:

\frac{1}{6}(x + 4) - \frac{5}{12}

.

\frac{1}{6}x + \frac{1}{4}

By distributing

\frac{1}{6}

, we get

\frac{1}{6}x + \frac{4}{6} - \frac{5}{12}

. Finding a common denominator of 12 for the constant terms yields

\frac{8}{12} - \frac{5}{12} = \frac{3}{12} = \frac{1}{4}

.

4

Equate the simplified left and right sides:

\frac{1}{2}x - \frac{1}{12} = \frac{1}{6}x + \frac{1}{4}

. Multiply both sides of the equation by 12 to clear the fractions.

6x - 1 = 2x + 3

, which simplifies to

4x = 4

, and thus

x = 1

.

To solve for the variable

x

in a simplified integer form.

5

Substitute

x = 1

into the requested expression

12x - 5

.

12(1) - 5 = 7

To calculate the final value asked by the question.

Key Concept

Solving multi-step linear equations in one variable with fractional coefficients, distributing negative signs, and evaluating expressions.

Question 32Question →

For all $x > 0$ , the expression $\frac{\sqrt{x^7}}{\sqrt[5]{x^3}}$ is equivalent to $x^a$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 2.9

Answer

2.9

To rewrite the expression, we convert the radicals to rational exponents:

\sqrt{x^7} = x^{7/2}

and

\sqrt[5]{x^3} = x^{3/5}

. Using the quotient property of exponents,

\frac{x^m}{x^n} = x^{m-n}

, we subtract the denominator's exponent from the numerator's exponent:

\frac{7}{2} - \frac{3}{5}

. Finding a common denominator of 10 gives

\frac{35}{10} - \frac{6}{10} = \frac{29}{10}

, which simplifies to

2.9

. Thus, the value of

a

is

2.9

.

Step-by-Step Solution

1

Convert the radical expressions to rational exponents.

Numerator:

\sqrt{x^7} = x^{7/2}

; Denominator:

\sqrt[5]{x^3} = x^{3/5}

.

Radical expressions

\sqrt[n]{x^m}

can be rewritten as rational exponent expressions

x^{m/n}

.

2

Apply the quotient rule of exponents.

x^{7/2} / x^{3/5} = x^{7/2 - 3/5}

.

When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Subtract the exponents by finding a common denominator.

\frac{7}{2} - \frac{3}{5} = \frac{35}{10} - \frac{6}{10} = \frac{29}{10} = 2.9

.

Subtracting the fractions requires a common denominator of 10. The result

\frac{29}{10}

is equivalent to the terminating decimal

2.9

.

Key Concept

Equivalent algebraic expressions involving radical and rational exponents

Question 33Question →

In 2024, a solar power system generated $250$ megawatt-hours of electricity. In 2025, the system's electricity generation was $12\%$ greater than in 2024. In 2026, the electricity generation was $5\%$ less than in 2025. How many megawatt-hours of electricity did the solar power system generate in 2026?

Show answer & explanation

Answer: 266

Answer

266

To find the electricity generation in 2026, we first determine the generation in 2025. Since the 2025 generation was

12\%

greater than the 2024 generation of

250

megawatt-hours, the 2025 generation is

250 \times (1 + 0.12) = 250 \times 1.12 = 280

megawatt-hours. Next, the 2026 generation was

5\%

less than the 2025 generation of

280

megawatt-hours. Therefore, the 2026 generation is

280 \times (1 - 0.05) = 280 \times 0.95 = 266

megawatt-hours.

Step-by-Step Solution

1

Calculate the electricity generation in 2025.

280 megawatt-hours

The 2025 generation is

12\%

greater than the 2024 generation of

250

megawatt-hours. So, we multiply

250

by

1 + 0.12 = 1.12

to get

280

.

2

Calculate the electricity generation in 2026.

266 megawatt-hours

The 2026 generation is

5\%

less than the 2025 generation of

280

megawatt-hours. So, we multiply

280

by

1 - 0.05 = 0.95

to get

266

.

Key Concept

Calculating consecutive percent changes by applying each percentage change to the correct sequential base value.

Alternative Method

We can set up a single equation representing the sequential changes:

\text{2026 Generation} = 250 \times (1 + 0.12) \times (1 - 0.05) = 250 \times 1.12 \times 0.95 = 266

megawatt-hours.

Estimated Time:1m 15s

Question 34Question →

A community garden charges a one-time registration fee plus a monthly fee to plot and maintain a garden bed. The total cost, $y$ , in dollars, of maintaining a garden bed for $x$ months is given by the equation $y = 15x + 45$ . If a gardener spent a total of $165$ dollars, for how many months did they maintain the garden bed?

Show answer & explanation

Answer: 8

Answer

The correct answer is 8. Setting the total cost y equal to 165 in the linear equation yields 165 = 15x + 45. Subtracting 45 from both sides gives 120 = 15x, and dividing by 15 results in x = 8 months.

To find the number of months the gardener maintained the garden bed, substitute the total spent,

165

, for

y

in the equation

y = 15x + 45

. This results in

165 = 15x + 45

. Subtracting

45

from both sides of the equation yields

120 = 15x

. Dividing both sides by

15

gives

x = 8

. Thus, the garden bed was maintained for 8 months.

Step-by-Step Solution

1

Substitute 165 for y in the equation.

165 = 15x + 45

The variable y represents the total cost in dollars, which is given as 165.

2

Subtract 45 from both sides of the equation.

120 = 15x

This isolates the variable term by subtracting the constant registration fee.

3

Divide both sides of the equation by 15.

x = 8

This solves for x, the number of months.

Key Concept

Solving for a variable in a linear equation representing a real-world scenario.

Question 35Question →

For all $x > 0$ , the expression $\frac{6x^2 + 13x - 5}{2x + 5} - \frac{3x^2 - 4x}{x}$ is equivalent to a constant $c$ . What is the value of $c$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

To find the constant value of the expression, both rational expressions are first simplified. The quadratic in the numerator of the first fraction,

6x^2 + 13x - 5

, can be factored as

(3x - 1)(2x + 5)

. Canceling the common factor of

2x + 5

in the numerator and denominator simplifies the first term to

3x - 1

. The second fraction,

\frac{3x^2 - 4x}{x}

, can be simplified by dividing each term in the numerator by

x

, which results in

3x - 4

. Subtracting the second simplified expression from the first gives

(3x - 1) - (3x - 4) = 3x - 1 - 3x + 4 = 3

. Thus, the constant value of the expression is

3

.

Step-by-Step Solution

1

Factor the numerator of the first rational expression.

6x^2 + 13x - 5 = (3x - 1)(2x + 5)

This allows us to simplify the fraction by canceling common factors.

2

Simplify the first fraction by canceling the common term

(2x + 5)

.

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

Since

x > 0

,

2x + 5 \neq 0

, so we can cancel the common factor.

3

Simplify the second fraction by factoring out

x

and canceling it.

\frac{3x^2 - 4x}{x} = 3x - 4

Since

x > 0

,

x \neq 0

, so we can divide each term in the numerator by

x

.

4

Subtract the two simplified expressions.

(3x - 1) - (3x - 4) = 3

To find the constant value

c

, we subtract the second simplified expression from the first and combine like terms, paying close attention to distributing the negative sign.

Key Concept

Simplifying rational expressions by factoring and performing polynomial subtraction.

Question 36Question →

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

x - y = 4

x^2 - y^2 = 40

Show answer & explanation

Answer: 7

Answer

The value of

x

is 7.

The equation

x^2 - y^2 = 40

can be factored as

(x-y)(x+y) = 40

. Substituting

x-y = 4

yields

4(x+y) = 40

, which simplifies to

x+y = 10

. Adding the two linear equations

x-y = 4

and

x+y = 10

eliminates

y

, resulting in

2x = 14

, or

x = 7

.

Step-by-Step Solution

1

Factor the second equation using the difference of squares identity.

(x-y)(x+y) = 40

To rewrite the quadratic expression in a form where the linear equation can be substituted.

2

Substitute the first equation

x-y = 4

into the factored expression.

4(x+y) = 40

, which simplifies to

x+y = 10

To find a simpler linear relation for the sum of the variables.

3

Add the equations

x-y = 4

and

x+y = 10

.

2x = 14

To eliminate the variable

y

and solve for

x

directly.

4

Solve for

x

by dividing by 2.

x = 7

To isolate the variable and find the final value of

x

.

Key Concept

Solving a nonlinear system of equations by factoring a difference of squares and substituting.

Alternative Method

Express

x

from the first equation as

x = y + 4

and substitute it into the second equation:

(y+4)^2 - y^2 = 40

. Expanding and simplifying yields

y^2 + 8y + 16 - y^2 = 40

, which simplifies to

8y + 16 = 40 \implies 8y = 24 \implies y = 3

. Substituting

y = 3

back into

x = y + 4

gives

x = 7

.

Estimated Time:1m 15s

Question 37Question →

An online bookstore charges a flat shipping fee plus a fixed price per book purchased. The total cost, $y$ , in dollars, for purchasing $x$ books is given by the equation $y = 8.5x + 4.5$ . If a customer's total cost was $47$ dollars, how many books did the customer purchase?

Show answer & explanation

Answer: 5

Answer

The customer purchased 5 books.

To find the number of books purchased, substitute the total cost of 47 for

y

in the equation

y = 8.5x + 4.5

, yielding

47 = 8.5x + 4.5

. Subtracting

4.5

from both sides gives

42.5 = 8.5x

. Dividing both sides by

8.5

results in

x = 5

. Thus, the customer purchased 5 books.

Step-by-Step Solution

1

Substitute the total cost of 47 into the equation for y

47 = 8.5x + 4.5

Since the total cost is represented by y and is given as 47 dollars, we substitute 47 for y in the equation.

2

Subtract 4.5 from both sides of the equation

42.5 = 8.5x

Subtracting the flat fee from both sides isolates the variable term representing the total cost of the books.

3

Divide both sides by 8.5

x = 5

Dividing the remaining cost by the price per book yields the total number of books purchased.

Key Concept

Linear Equations in Two Variables

Question 38Question →

If $4^{2n-3} = \left(\frac{1}{8}\right)^{n-5}$ , what is the value of $n$ ?

Show answer & explanation

Answer: 3

Answer

3

Rewriting the bases 4 and

\frac{1}{8}

as powers of 2 gives

(2^2)^{2n-3} = (2^{-3})^{n-5}

. Applying the exponent power rule simplifies this to

2^{4n-6} = 2^{-3n+15}

. Since the bases are equal, the exponents must be equal:

4n-6 = -3n+15

. Solving this linear equation by adding

3n

and 6 to both sides gives

7n = 21

, which yields

n = 3

.

Step-by-Step Solution

1

Express the bases 4 and

\frac{1}{8}

as powers of the prime base 2.

4 = 2^2

and

\frac{1}{8} = 2^{-3}

Rewriting the bases with a common base of 2 allows for the application of exponent rules to solve the equation.

2

Substitute these bases back into the original equation.

(2^2)^{2n-3} = (2^{-3})^{n-5}

This sets up the equation to simplify the exponent terms.

3

Apply the power of a power rule,

(a^b)^c = a^{bc}

, by multiplying the exponents on both sides.

2^{2(2n-3)} = 2^{-3(n-5)} \implies 2^{4n-6} = 2^{-3n+15}

Multiplying the exponents simplifies the expressions to a single base raised to a single exponent on each side.

4

Equate the exponents since the bases are now equal, and solve for

n

.

4n-6 = -3n+15 \implies 7n = 21 \implies n = 3

If

b^x = b^y

where

b > 0

and

b \neq 1

, then

x = y

.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and applying exponent rules.

Question 39Question →

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

y = 2x + 1

3x + 2y = 16

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Substituting

y = 2x + 1

into the second equation yields

3x + 2(2x + 1) = 16

. Expanding this expression gives

3x + 4x + 2 = 16

, which simplifies to

7x + 2 = 16

. Subtracting 2 from both sides gives

7x = 14

, and dividing by 7 yields

x = 2

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

3x + 2(2x + 1) = 16

This reduces the system of two variables to a single equation in terms of

x

.

2

Expand and simplify the equation by distributing and combining like terms.

7x + 2 = 16

Simplifying the equation makes it easier to isolate the variable

x

.

3

Isolate the variable

x

by performing inverse operations.

x = 2

Subtracting 2 and then dividing by 7 isolates

x

to find its value.

Key Concept

Solving a system of linear equations using the substitution method.

Alternative Method

Alternatively, you can multiply the first equation by 2 to get

-4x + 2y = 2

and subtract it from

3x + 2y = 16

to eliminate

y

, which also yields

7x = 14

and thus

x = 2

.

Estimated Time:1m 0s

Question 40Question →

A line representing the linear function $f$ is graphed in the $xy$ -plane. The line intersects the $y$ -axis at $(0, 3)$ and also passes through the point $(2, 11)$ . What is the slope of this line?

Show answer & explanation

Answer: 4

Answer

The slope of the line is 4.

The slope

m

of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

. Substituting the given points

(0, 3)

and

(2, 11)

into the formula gives

m = \frac{11 - 3}{2 - 0} = \frac{8}{2} = 4

.

Step-by-Step Solution

1

Identify two coordinates on the line from the given information.

The points are

(0, 3)

and

(2, 11)

.

To find the slope of a line, we need at least two points on that line.

2

Apply the slope formula

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

.

m = \frac{11 - 3}{2 - 0}

The slope formula calculates the ratio of the vertical change (rise) to the horizontal change (run).

3

Perform the subtraction and division to find the slope value.

m = \frac{8}{2} = 4

Simplifying the fraction gives the final numerical slope.

Key Concept

Calculating the slope of a linear function given two points

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