All practice questions

Question 161Question →

If the expression $\frac{(x^2 y^3)^4}{(x^{-1} y^2)^3}$ is written in the equivalent form $x^a y^b$ , what is the value of $a - b$ ?

Show answer & explanation

Answer: 5

Answer

5

Simplifying the numerator yields

x^8 y^{12}

and simplifying the denominator yields

x^{-3} y^6

. Dividing these expressions by subtracting the exponents of like bases results in

x^{8 - (-3)} y^{12 - 6} = x^{11} y^6

. Comparing this to the expression

x^a y^b

shows that

a = 11

and

b = 6

. The value of

a - b

is

11 - 6 = 5

.

Step-by-Step Solution

1

Simplify the numerator of the expression.

x^8 y^{12}

Apply the power of a power and power of a product properties of exponents:

(x^2 y^3)^4 = x^{2 \cdot 4} y^{3 \cdot 4}

.

2

Simplify the denominator of the expression.

x^{-3} y^6

Apply the power of a power and power of a product properties of exponents:

(x^{-1} y^2)^3 = x^{-1 \cdot 3} y^{2 \cdot 3}

.

3

Simplify the quotient by dividing the simplified numerator by the simplified denominator.

x^{11} y^6

Use the quotient property of exponents,

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

, to subtract the exponents:

8 - (-3) = 11

and

12 - 6 = 6

.

4

Identify the values of

a

and

b

and compute the difference

a - b

.

5

By comparing

x^{11} y^6

to the target form

x^a y^b

, we find

a = 11

and

b = 6

. Subtracting

b

from

a

yields

11 - 6 = 5

.

Key Concept

Properties of exponents including power of a power, power of a product, and quotient rules.

Question 162Question →

A parabola is defined by the equation $y = 2x^2 - 5x + 1$ and a line is defined by the equation $y = x - 3$ . If the parabola and the line intersect at the points $(x_1, y_1)$ and $(x_2, y_2)$ , what is the value of $y_1 + y_2$ ?

Show answer & explanation

Answer: -3

Answer

The sum of the y-coordinates of the intersection points is -3.

By setting the two equations equal to each other, we obtain a quadratic equation in terms of

x

. Solving this equation gives the x-coordinates of the intersection points. Substituting these x-values back into the linear equation yields the corresponding y-coordinates. Summing these y-coordinates gives the final value of

-3

.

Step-by-Step Solution

1

Equate the equations for the line and the parabola to find the x-coordinates of their intersection points.

2x^2 - 5x + 1 = x - 3

Since both equations are equal to

y

, their right-hand sides must be equal at the points of intersection.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

2x^2 - 6x + 4 = 0

Grouping all terms on one side of the equation allows us to solve for $x.

3

Divide the entire equation by 2 to simplify.

x^2 - 3x + 2 = 0

Simplifying the quadratic equation makes factoring easier.

4

Factor the quadratic equation to solve for

x

.

(x - 1)(x - 2) = 0

, yielding

x = 1

and

x = 2

.

Finding the roots of the quadratic equation gives the x-coordinates of the intersection points.

5

Substitute each x-coordinate back into the linear equation

y = x - 3

to find the corresponding y-coordinates.

For

x_1 = 1

:

y_1 = 1 - 3 = -2

. For

x_2 = 2

:

y_2 = 2 - 3 = -1

.

The intersection points must satisfy both equations in the system.

6

Calculate the sum of the two y-coordinates,

y_1 + y_2

.

y_1 + y_2 = -2 + (-1) = -3

The question asks for the value of the sum of the y-coordinates.

Key Concept

Solving systems of linear and quadratic equations by substitution

Question 163Question →

Two polynomial expressions are defined as $P(x) = (2x - 4)(x^2 - 5x + c)$ and $Q(x) = (x^2 + 3x - 4)(2x + a)$ , where $a$ and $c$ are constants. When $P(x)$ is expanded and simplified, it has no $x$ term. If the coefficient of the $x^2$ term in the expanded and simplified form of $Q(x)$ is equal to the coefficient of the $x^2$ term in the expanded and simplified form of $P(x)$ , what is the value of $a$ ?

Show answer & explanation

Answer: -20

Answer

The value of

a

is

-20

.

Expanding

P(x) = (2x - 4)(x^2 - 5x + c)

yields

2x^3 - 14x^2 + (2c + 20)x - 4c

. Since there is no

x

term,

2c + 20 = 0

, which means

c = -10

. This leaves the coefficient of the

x^2

term in

P(x)

as

-14

. Expanding

Q(x) = (x^2 + 3x - 4)(2x + a)

yields

2x^3 + (a + 6)x^2 + (3a - 8)x - 4a

, so the coefficient of its

x^2

term is

a + 6

. Setting

a + 6 = -14

and solving for

a

gives

a = -20

.

Step-by-Step Solution

1

Expand the polynomial

P(x) = (2x - 4)(x^2 - 5x + c)

using the distributive property.

P(x) = 2x^3 - 14x^2 + (2c + 20)x - 4c

To identify the coefficients of each term in

P(x)

.

2

Set the coefficient of the

x

term,

2c + 20

, to

0

and solve for

c

.

c = -10

The problem states that

P(x)

has no

x

term when simplified, which means its coefficient must be zero.

3

Determine the coefficient of the

x^2

term in

P(x)

.

The coefficient of

x^2

is

-14

.

This coefficient will be equated to the

x^2

coefficient of

Q(x)

as per the problem constraints.

4

Expand the polynomial

Q(x) = (x^2 + 3x - 4)(2x + a)

using the distributive property.

Q(x) = 2x^3 + (a + 6)x^2 + (3a - 8)x - 4a

To identify the coefficient of the

x^2

term in

Q(x)

.

5

Set the coefficient of the

x^2

term in

Q(x)

, which is

a + 6

, equal to the coefficient of the

x^2

term in

P(x)

, which is

-14

, and solve for

a

.

a = -20

The problem states that these two coefficients are equal.

Key Concept

Operations on Polynomials

Estimated Time:2m 30s

Question 164Question →

A construction company is working on two building projects, Project X and Project Y. The quantities of concrete (in tons) and steel (in tons) required for these projects are represented by the matrix $Q$ :

Q = \begin{bmatrix} 80 & 25 \\ 120 & 40 \end{bmatrix}

where the first row represents Project X, the second row represents Project Y, the first column represents concrete, and the second column represents steel.

The purchase cost per ton and the transportation cost per ton for these materials are represented by the matrix $C$ :

C = \begin{bmatrix} 110 & 15 \\ 650 & 45 \end{bmatrix}

where the first row represents concrete, the second row represents steel, the first column represents the purchase cost (in dollars per ton), and the second column represents the transportation cost (in dollars per ton).

If the product matrix $P = QC$ represents the total cost details for the two projects, what is the total transportation cost for Project Y, in dollars?

Show answer & explanation

Answer: 3600

Answer

The total transportation cost for Project Y is 3,600 dollars.

The correct answer is 3,600. To find the total transportation cost for Project Y, we calculate the entry in the second row (Project Y) and second column (transportation cost) of the product matrix

P = QC

. This is calculated as

120 \text{ tons of concrete} \times 15 \text{ dollars per ton} + 40 \text{ tons of steel} \times 45 \text{ dollars per ton} = 1,800 + 1,800 = 3,600

dollars.

Step-by-Step Solution

1

Identify the row representing Project Y and the column representing transportation cost

Row 2 of

Q

is

[120, 40]

and Column 2 of

C

is

[15, 45]^T

.

To find the total transportation cost for Project Y, we must compute the entry in the second row (Project Y) and second column (transportation cost) of the product matrix

P = QC

.

2

Multiply the row elements by the corresponding column elements

120 \times 15 = 1800

and

40 \times 45 = 1800

This calculates the individual transportation costs for the concrete and steel required for Project Y.

3

Sum the products to find the total transportation cost

1800 + 1800 = 3600

Adding these individual costs gives the total transportation cost for Project Y.

Key Concept

Matrix multiplication involves multiplying the elements of each row of the first matrix by the corresponding elements of each column of the second matrix and summing the products.

Question 165Question →

If $3^{2x - 1} = 27$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is

2

.

Rewriting

27

as

3^3

gives the equation

3^{2x - 1} = 3^3

. Equating the exponents yields

2x - 1 = 3

, which solves to

x = 2

.

Step-by-Step Solution

1

Rewrite the right side of the equation with a base of

3

.

3^{2x - 1} = 3^3

Expressing both sides of the equation with a common base allows for direct comparison of the exponents.

2

Set the exponents equal to each other.

2x - 1 = 3

Since the bases are both

3

, their exponents must be equal for the expressions to be equal.

3

Solve the linear equation for

x

.

x = 2

Adding

1

to both sides gives

2x = 4

. Dividing both sides by

2

results in

x = 2

.

Key Concept

Solving exponential equations by expressing both sides with a common base

Question 166Question →

If $x$ is a real number such that $\frac{(3^x)^4 \cdot 9^{2-x}}{27^{x-1}} = 81$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Rewriting the bases in terms of 3, the expression becomes

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

. Combining the numerator using the product rule gives

3^{2x+4}

in the numerator. Dividing by the denominator using the quotient rule gives

3^{(2x+4)-(3x-3)} = 3^{-x+7}

. Equating this to

3^4

gives

-x + 7 = 4

, which yields

x = 3

.

Step-by-Step Solution

1

Express each base in terms of the common base 3

9 = 3^2

,

27 = 3^3

, and

81 = 3^4

Converting all terms to a common base allows the use of exponent rules to combine them.

2

Apply the power of a power rule

(b^m)^n = b^{mn}

to rewrite each exponent

(3^x)^4 = 3^{4x}

,

(3^2)^{2-x} = 3^{4-2x}

, and

(3^3)^{x-1} = 3^{3x-3}

This simplifies individual terms by multiplying their exponents.

3

Apply the product rule of exponents

b^m \cdot b^n = b^{m+n}

to combine the terms in the numerator

3^{4x} \cdot 3^{4-2x} = 3^{4x + 4 - 2x} = 3^{2x + 4}

Multiplying exponential terms with the same base is simplified by adding their exponents.

4

Apply the quotient rule of exponents

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

to simplify the fraction

3^{(2x+4) - (3x-3)} = 3^{-x+7}

Dividing exponential terms with the same base is simplified by subtracting the exponent in the denominator from the exponent in the numerator.

5

Equate the exponents of the simplified base 3 expression and base 3 representation of 81

-x + 7 = 4

, which solves to

x = 3

Since the bases are equal, the powers must be equal for the equation to hold true.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 167Question →

A line intersects a parabola at two distinct points in the standard $(x, y)$ coordinate plane. The system of equations representing these curves is given by:

\begin{aligned} x^2 - 7x + 3y &= 6 \\ 3y - 4x &= 6 \end{aligned}

What is the distance between the two intersection points?

Show answer & explanation

Answer: 5

Answer

5

Solving the system of equations yields the intersection points

(0, 2)

and

(3, 6)

. The distance between these points is computed using the distance formula:

\sqrt{(3 - 0)^2 + (6 - 2)^2} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Isolate the

3y

term in the linear equation.

3y = 4x + 6

This allows for a direct substitution into the quadratic equation without introducing fractional coefficients.

2

Substitute

3y

into the first equation and simplify.

x^2 - 3x = 0

Substituting

4x + 6

for

3y

yields

x^2 - 7x + (4x + 6) = 6

. Subtracting 6 from both sides and combining like terms simplifies this to a basic quadratic equation.

3

Solve the quadratic equation for

x

and determine the corresponding

y

-coordinates.

The intersection points are

(0, 2)

and

(3, 6)

.

Factoring gives

x(x - 3) = 0

, so the

x

-coordinates are

0

and

3

. Substituting

x = 0

into the linear relation gives

3y = 6 \implies y = 2

. Substituting

x = 3

gives

3y = 18 \implies y = 6

.

4

Calculate the distance between the two coordinates.

5

Using the distance formula:

d = \sqrt{(3 - 0)^2 + (6 - 2)^2} = \sqrt{9 + 16} = 5

.

Key Concept

Systems of Linear and Non-Linear Equations

Question 168Question →

What is the greatest integer value of $x$ that satisfies the inequality $\frac{2 - 5x}{3} - \frac{x - 4}{2} \geq 3$ ?

Show answer & explanation

Answer: -1

Answer

The greatest integer value of

x

that satisfies the inequality is -1.

The correct answer is -1 because solving the inequality leads to

x \leq -\frac{2}{13}

. Since

-\frac{2}{13}

is approximately

-0.154

, the set of integers satisfying the inequality is

\{-1, -2, -3, \dots\}

. The greatest integer in this set is -1.

Step-by-Step Solution

1

Multiply the entire inequality by the least common multiple of the denominators (6) to eliminate the fractions.

2(2 - 5x) - 3(x - 4) \geq 18

Multiplying by a positive number clears the fractions without changing the direction of the inequality.

2

Distribute the coefficients and combine like terms on the left side of the inequality.

16 - 13x \geq 18

Simplifying the expressions on each side makes it easier to isolate the variable.

3

Subtract 16 from both sides to isolate the term with the variable

x

.

-13x \geq 2

Moving the constant terms to one side prepares the inequality for division.

4

Divide both sides by -13 and reverse the direction of the inequality sign.

x \leq -\frac{2}{13}

Dividing by a negative number requires flipping the inequality sign to maintain a true statement.

5

Determine the greatest integer that is less than or equal to

-\frac{2}{13}

.

-1

Since

-\frac{2}{13} \approx -0.154

, the largest integer that is less than or equal to this value is -1.

Key Concept

Solving multi-step linear inequalities, including clearing fractional coefficients and reversing the inequality sign when multiplying or dividing by a negative number.

Question 169Question →

What is the value of the discriminant of the quadratic equation $3x^2 + 5x - 2 = 0$ ?

Show answer & explanation

Answer: 49

Answer

The discriminant of the quadratic equation is 49.

The discriminant is calculated using the formula

b^2 - 4ac

. For the equation

3x^2 + 5x - 2 = 0

, the coefficients are

a = 3

,

b = 5

, and

c = -2

. Substituting these yields

5^2 - 4(3)(-2) = 25 - (-24) = 25 + 24 = 49

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

3x^2 + 5x - 2 = 0

in the standard form

ax^2 + bx + c = 0

.

a = 3

,

b = 5

, and

c = -2

To use the discriminant formula, we must first extract the constant coefficients corresponding to each term.

2

Substitute the coefficients into the discriminant formula

D = b^2 - 4ac

.

D = 5^2 - 4(3)(-2) = 25 - (-24) = 25 + 24 = 49

Calculating the value of the discriminant provides the required solution.

Key Concept

Calculating the discriminant of a quadratic equation to determine the nature of its roots.

Question 170Question →

What is the maximum integer value of $k$ that satisfies the inequality $8 - 5k > 28$ ?

Show answer & explanation

Answer: -5

Answer

The maximum integer value that satisfies the inequality is

-5

.

Subtracting 8 from both sides of the inequality

8 - 5k > 28

yields

-5k > 20

. Dividing both sides of the inequality by

-5

and reversing the inequality sign results in

k < -4

. The largest integer strictly less than

-4

is

-5

.

Step-by-Step Solution

1

Subtract 8 from both sides of the inequality.

-5k > 20

To isolate the term with the variable on the left side of the inequality.

2

Divide both sides by

-5

and reverse the inequality sign.

k < -4

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality symbol.

3

Determine the largest integer strictly less than

-4

.

-5

Because the inequality is strict (

<

), the value of

k

cannot be equal to

-4

. The greatest integer less than

-4

is

-5

.

Key Concept

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.

Question 171Question →

Consider the functions $f(x) = \frac{x+3}{x-1}$ , where $x \neq 1$ , and $g(x) = x^2 - x - 4$ . If $x$ is an integer such that the composite function evaluation $g(f(g(x))) = 16$ , what is the product of all such integer values of $x$ ?

Show answer & explanation

Answer: -6

Answer

The product of all integer values of

x

that satisfy the equation is

-6

.

Solving the nested composite equation

g(f(g(x))) = 16

systematically yields the integer solutions

x = 3

and

x = -2

. The product of these values is

-6

.

Step-by-Step Solution

1

Set

u = f(g(x))

and solve the outer quadratic equation

g(u) = 16

.

u = 5

or

u = -4

This simplifies the nested composition into a single-variable quadratic equation.

2

Set

v = g(x)

and solve the rational equations

f(v) = 5

and

f(v) = -4

.

v = 2

or

v = 0.2

This determines the required outputs of the function

g(x)

that will satisfy the composite equation.

3

Solve the quadratic equations

g(x) = 2

and

g(x) = 0.2

for

x

.

x = 3

,

x = -2

, or

x = \frac{5 \pm \sqrt{445}}{10}

This finds all real values of

x

that satisfy the composite equation.

4

Identify the integer values from the solution set and compute their product.

3 \times (-2) = -6

The question specifically requests the product of the integer values of

x

.

Key Concept

Function composition and multi-step equation solving

Question 172Question →

For a constant $k$ , the quadratic equation $\frac{1}{3}x^2 - \frac{k}{6}x + (k - 10) = 0$ has two real roots, $r_1$ and $r_2$ . If the sum of the reciprocals of the roots, $\frac{1}{r_1} + \frac{1}{r_2}$ , is equal to $-\frac{1}{2}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7.5

Answer

The value of

k

is

7.5

.

By writing the sum of the reciprocals in terms of the sum and product of the roots, we find that

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

. Substituting the values from Vieta's formulas (

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

) yields

\frac{k}{6(k - 10)} = -\frac{1}{2}

. Solving this equation for

k

results in

k = 7.5

.

Step-by-Step Solution

1

Identify the quadratic coefficients in terms of the constant

k

.

a = \frac{1}{3}

,

b = -\frac{k}{6}

, and

c = k - 10

Applying Vieta's formulas requires the coefficients from the standard form

ax^2 + bx + c = 0

.

2

Determine the sum and the product of the roots using Vieta's formulas.

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

Vieta's formulas state that the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

3

Express the sum of the reciprocals of the roots in terms of

k

.

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2} = \frac{k}{6(k - 10)}

Finding a common denominator allows the sum of the reciprocals to be written as the ratio of the sum of the roots to the product of the roots.

4

Equate the expression to

-\frac{1}{2}

and solve the resulting equation for

k

.

k = 7.5

Solving

\frac{k}{6(k - 10)} = -\frac{1}{2}

gives

2k = -6(k - 10) \implies 8k = 60 \implies k = 7.5

.

Key Concept

Vieta's Formulas and Algebraic Relationships of Roots

Estimated Time:2m 30s

Question 173Question →

For the opening night of a school play, a total of $320$ tickets were sold, raising a total of $\$2{}140$ in ticket sales. Student tickets were sold for $\$5$ each, and adult tickets were sold for $\$8$ each. How many adult tickets were sold?

Show answer & explanation

Answer: 180

Answer

180

The correct answer is

180

. By setting up a single-variable linear equation where

a

represents the number of adult tickets, the problem translates to

8a + 5(320 - a) = 2140

. Simplifying this yields

3a + 1600 = 2140

, which simplifies to

3a = 540

and results in

a = 180

.

Step-by-Step Solution

1

Define variables for the unknown quantities based on the given total.

Let

a

represent the number of adult tickets sold. The number of student tickets sold is represented as

320 - a

.

Since the total number of tickets is

320

, subtracting the number of adult tickets from the total yields the number of student tickets.

2

Construct a linear equation using the ticket prices and the total revenue.

8a + 5(320 - a) = 2140

Multiplying the quantity of each ticket type by its respective price (

8

dollars for adult and

5

dollars for student) yields the total ticket sales revenue of

2,140

dollars.

3

Solve the equation for the variable

a

.

8a + 1600 - 5a = 2140 \implies 3a + 1600 = 2140 \implies 3a = 540 \implies a = 180

Apply the distributive property, group like terms, isolate the variable term, and divide to solve for the number of adult tickets.

Key Concept

Translating word problems into a single-variable linear equation and solving for the unknown quantity.

Question 174Question →

The table below shows some values of the functions $f$ and $g$ for several integer values of $x$ .

$x$	$f(x)$	$g(x)$
$-2$	$3$	$1$
$-1$	$4$	$-2$
$0$	$-1$	$3$
$1$	$2$	$0$
$2$	$-2$	$-1$
$3$	$1$	$2$

What is the value of $f(f(3)) + g(g(-1))$ ?

Show answer & explanation

Answer: 3

Answer

The value of the expression is

3

.

To evaluate

f(f(3)) + g(g(-1))

, we start by evaluating the innermost expressions. Looking at the table for

x = 3

, we find

f(3) = 1

. Looking at the table for

x = -1

, we find

g(-1) = -2

. Substituting these values into the outer functions gives

f(1) + g(-2)

. Using the table again, we look up

x = 1

to find

f(1) = 2

, and look up

x = -2

to find

g(-2) = 1

. Adding these two results yields

2 + 1 = 3

.

Step-by-Step Solution

1

Find the inner function values from the given table.

f(3) = 1

and

g(-1) = -2

Before evaluating the composite functions, we must find the outputs of the innermost functions.

2

Evaluate the outer functions using the results from the first step.

f(f(3)) = f(1) = 2

and

g(g(-1)) = g(-2) = 1

Substitute the inner outputs into the outer functions and look up the new inputs in the table.

3

Add the two resulting values together.

2 + 1 = 3

Combine the two terms to find the total sum requested by the question.

Key Concept

Evaluating composite functions using tables of values

Estimated Time:1m 0s

Question 175Question →

A company sells two types of gift baskets: Standard and Premium. The sales at the company's North and South branches are represented by matrix $S$ :

S = \begin{bmatrix} 12 & 8 \\ 15 & 10 \end{bmatrix}

where the rows represent the North and South branches, respectively, and the columns represent the number of Standard and Premium baskets sold, respectively. The cost of the items inside each basket is represented by matrix $C$ :

C = \begin{bmatrix} x & 24 \\ 30 & y \end{bmatrix}

where the rows represent Standard and Premium baskets, respectively, and the columns represent the cost of food items and the cost of gift items (in dollars), respectively. If the total food item cost at the North branch is $600$ and the total gift item cost at the South branch is $760$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 70

Answer

The value of

x + y

is 70.

To find the total costs, the sales matrix

S

is multiplied by the unit cost matrix

C

. The product matrix

R = SC

is a

2 \times 2

matrix where the entry in Row 1, Column 1 represents the total food cost at the North branch, and the entry in Row 2, Column 2 represents the total gift cost at the South branch. Setting up the equations

12x + 240 = 600

and

360 + 10y = 760

yields

x = 30

and

y = 40

, which sum to 70.

Step-by-Step Solution

1

Set up the matrix multiplication

R = SC

to represent the total costs.

R = \begin{bmatrix} 12x + 240 & 288 + 8y \\ 15x + 300 & 360 + 10y \end{bmatrix}

The product of the sales matrix and the unit cost matrix yields the total cost matrix, where rows represent the branches and columns represent the cost categories.

2

Identify the expression for the total food item cost at the North branch and solve for

x

.

x = 30

The total food item cost at the North branch is the entry in Row 1, Column 1 of the product matrix, which is

12x + 8(30) = 12x + 240

. Setting this equal to

600

gives

12x + 240 = 600

, which simplifies to

12x = 360

, so

x = 30

.

3

Identify the expression for the total gift item cost at the South branch and solve for

y

.

y = 40

The total gift item cost at the South branch is the entry in Row 2, Column 2 of the product matrix, which is

15(24) + 10y = 360 + 10y

. Setting this equal to

760

gives

360 + 10y = 760

, which simplifies to

10y = 400

, so

y = 40

.

4

Calculate the sum of the variables

x

and

y

.

x + y = 70

Substitute

x = 30

and

y = 40

into the expression

x + y

to find the final answer.

Key Concept

Matrix multiplication and translating real-world scenarios into matrix equations.

Estimated Time:3m 0s

Question 176Question →

In the standard $(x,y)$ coordinate plane, a line has the equation $3x + 2y = 12$ . What is the $x$ -intercept of this line?

Show answer & explanation

Answer: 4

Answer

The

x

-intercept of the line is

4

.

Step-by-Step Solution

1

Substitute

y = 0

into the equation to find the point where the line intersects the

x

-axis.

3x + 2(0) = 12

By definition, the

x

-intercept occurs where the

y

-coordinate is equal to

0

.

2

Simplify the equation and solve for the variable

x

.

3x = 12 \Rightarrow x = 4

Simplifying

2(0)

to

0

leaves

3x = 12

. Dividing both sides of the equation by

3

isolates

x

, giving

x = 4

.

Key Concept

Determining the

x

-intercept of a linear equation by evaluating it at

y = 0

.

Estimated Time:45s

Question 177Question →

If $\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}$ , what is the value of $3x + 2$ ?

Show answer & explanation

Answer: 20

Answer

The correct answer is 20, which is the value of the expression

3x + 2

when

x = 6

.

The value of the expression

3x + 2

is 20. Solving the linear equation

\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}

yields

x = 6

. Substituting

x = 6

into

3x + 2

gives

3(6) + 2 = 20

.

Step-by-Step Solution

1

Multiply the entire equation by the least common multiple of the denominators (12).

8(3x - 4) - 3(x + 2) = 88

This clears the fractions to make solving the equation simpler.

2

Distribute the coefficients to remove parentheses.

24x - 32 - 3x - 6 = 88

This allows like terms to be grouped together.

3

Combine like terms on the left side of the equation.

21x - 38 = 88

Simplifies the equation to prepare for isolating the variable.

4

Add 38 to both sides of the equation.

21x = 126

Isolates the variable term on one side of the equation.

5

Divide both sides by 21 to solve for

x

.

x = 6

Finds the value of the variable

x

.

6

Substitute

x = 6

into the target expression

3x + 2

.

3(6) + 2 = 20

Calculates the final requested value.

Key Concept

Solving multi-step linear equations with fractions and evaluating algebraic expressions.

Question 178Question →

For a certain real number $x$ , the equation $\frac{3}{4}(x - 3) - \frac{1}{3}(2x + 5) = -2$ is true. What is the value of $2x - 5$ ?

Show answer & explanation

Answer: 41

Answer

The value of the expression

2x - 5

is 41.

Solving the linear equation by multiplying both sides by the least common denominator of 12 yields the simplified equation

9(x - 3) - 4(2x + 5) = -24

. Expanding the terms gives

9x - 27 - 8x - 20 = -24

, which simplifies to

x - 47 = -24

. Adding 47 to both sides gives

x = 23

. Finally, evaluating the expression

2x - 5

for

x = 23

results in

2(23) - 5 = 41

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator of 12 to eliminate fractions.

9(x - 3) - 4(2x + 5) = -24

Multiplying by the LCD clears all fractional coefficients, making the equation easier to solve.

2

Distribute the constants and expand the terms on the left side of the equation.

9x - 27 - 8x - 20 = -24

Applying the distributive property removes the parentheses.

3

Combine like terms on the left side of the equation.

x - 47 = -24

Simplifying the equation makes it easier to isolate the variable

x

.

4

Isolate the variable

x

by adding 47 to both sides of the equation.

x = 23

This determines the value of the unknown variable

x

.

5

Substitute

x = 23

into the target expression

2x - 5

.

41

The question asks for the value of the expression

2x - 5

, not just the value of

x

.

Key Concept

Solving multi-step linear equations with fractional coefficients by clearing the denominators and then evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the LCD first, distribute the fractions directly:

\frac{3}{4}x - \frac{9}{4} - \frac{2}{3}x - \frac{5}{3} = -2

. Combine the

x

terms:

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

. Combine the constant terms:

-\frac{9}{4} - \frac{5}{3} = -\frac{27}{12} - \frac{20}{12} = -\frac{47}{12}

. This gives the equation

\frac{1}{12}x - \frac{47}{12} = -2

. Add

\frac{47}{12}

to both sides:

\frac{1}{12}x = -2 + \frac{47}{12} = -\frac{24}{12} + \frac{47}{12} = \frac{23}{12}

. Multiply by 12 to get

x = 23

, then evaluate

2x - 5 = 41

.

Estimated Time:1m 30s

Question 179Question →

Matrices $A$ and $B$ are defined as:

A = \begin{bmatrix} x & 4 \\ -3 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}

Let $C$ represent the product matrix $AB$ , where:

C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}

If the element $c_{12}$ is equal to $2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

5

To find the entry

c_{12}

in the first row and second column of the product matrix

C = AB

, we calculate the dot product of the first row of matrix

A

and the second column of matrix

B

. The first row of

A

is

[x, 4]

and the second column of

B

is

[-2, 3]^T

. Thus,

c_{12} = x(-2) + 4(3) = -2x + 12

. Setting this equal to the given value of

2

gives the equation

-2x + 12 = 2

. Subtracting

12

from both sides results in

-2x = -10

. Dividing by

-2

yields

x = 5

.

Step-by-Step Solution

1

Identify the formula for the element

c_{12}

in the product matrix

C = AB

.

c_{12} = -2x + 12

The element

c_{12}

is located in the first row and second column of the product matrix, so it is the product of the first row of

A

, which is

[x, 4]

, and the second column of

B

, which is

[-2, 3]^T

.

2

Set the expression for

c_{12}

equal to the given value of

2

and solve the linear equation for

x

.

x = 5

Setting

-2x + 12 = 2

leads to

-2x = -10

, and dividing both sides by

-2

yields

x = 5

.

Key Concept

Matrix Multiplication and Element-wise Operations

Question 180Question →

If $x$ is a positive real number such that $\log_2(x) + \log_4(x) + \log_{16}(x) = 7$ , what is the value of $x$ ?

Show answer & explanation

Answer: 16

Answer

The value of

x

is

16

.

By converting all logarithms to base 2, we write

\log_4(x)

as

\frac{1}{2}\log_2(x)

and

\log_{16}(x)

as

\frac{1}{4}\log_2(x)

. Summing these gives

\frac{7}{4}\log_2(x) = 7

, which simplifies to

\log_2(x) = 4

. Converting to exponential form, we find

x = 2^4 = 16

.

Step-by-Step Solution

1

Apply the change of base formula to express the logarithms with a common base of 2.

\log_4(x) = \frac{1}{2}\log_2(x)

and

\log_{16}(x) = \frac{1}{4}\log_2(x)

, giving the equation

\log_2(x) + \frac{1}{2}\log_2(x) + \frac{1}{4}\log_2(x) = 7

.

Rewriting the terms with a common base allows them to be combined algebraically.

2

Combine the coefficients of the like terms on the left-hand side.

\frac{7}{4}\log_2(x) = 7

.

The sum of the coefficients is

1 + \frac{1}{2} + \frac{1}{4} = \frac{4}{4} + \frac{2}{4} + \frac{1}{4} = \frac{7}{4}

.

3

Isolate the logarithm term by dividing or multiplying by the reciprocal coefficient.

\log_2(x) = 4

.

Multiplying both sides by

\frac{4}{7}

solves for the value of

\log_2(x)

.

4

Convert the equation from logarithmic form to its equivalent exponential form.

x = 2^4 = 16

.

By definition,

\log_b(a) = c

is equivalent to

b^c = a

.

Key Concept

Change of Base Formula for Logarithms

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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