Intermediate Algebra Practice Questions — ACT — Page 7

Question 121Question →

The equation of a circle is $x^2 + y^2 = 25$ , and the equation of a line is $2x - y = 5$ . If the line intersects the circle at the points $P$ and $Q$ , what is the sum of the $y$ -coordinates of $P$ and $Q$ ?

-18

-6

-2

2

4

Show answer & explanation

Answer: -2

Answer

The sum of the

y

-coordinates is

-2

.

Substituting the linear equation into the circle equation yields a quadratic equation with roots

x = 0

and

x = 4

. Evaluating the linear equation at these values gives the

y

-coordinates

-5

and

3

. Adding these

y

-coordinates results in

-2

.

Step-by-Step Solution

1

Rearrange the linear equation to express

y

in terms of

x

.

y = 2x - 5

Expressing one variable in terms of the other allows for substitution into the quadratic circle equation.

2

Substitute

y = 2x - 5

into the circle equation

x^2 + y^2 = 25

and simplify.

x^2 + (2x - 5)^2 = 25 \implies x^2 + 4x^2 - 20x + 25 = 25 \implies 5x^2 - 20x = 0

This substitution reduces the system of equations to a single quadratic equation in terms of

x

.

3

Solve the quadratic equation

5x^2 - 20x = 0

for

x

.

5x(x - 4) = 0 \implies x_1 = 0

and

x_2 = 4

Finding the roots of this quadratic equation gives the

x

-coordinates of the intersection points.

4

Substitute the

x

-values back into the linear equation

y = 2x - 5

to find the corresponding

y

-coordinates.

For

x_1 = 0

,

y_1 = 2(0) - 5 = -5

. For

x_2 = 4

,

y_2 = 2(4) - 5 = 3

. The intersection points are

P(0, -5)

and

Q(4, 3)

.

This step determines the coordinates of the two points of intersection.

5

Calculate the sum of the

y

-coordinates of the points

P

and

Q

.

y_1 + y_2 = -5 + 3 = -2

This yields the final value requested by the question.

Key Concept

Solving a system of linear and circular equations by substitution

Estimated Time:2m 0s

Question 122Question →

What is the value of the larger real solution to the equation $\frac{10}{x} - \frac{3}{x-1} = 2$ ?

Show answer & explanation

Answer: 2.5

Answer

The larger real solution is

2.5

.

Multiplying the equation by the least common denominator

x(x-1)

yields

10(x-1) - 3x = 2x(x-1)

. Simplifying this leads to

2x^2 - 9x + 10 = 0

. Factoring the quadratic yields

(2x-5)(x-2) = 0

, which gives the solutions

x = 2.5

and

x = 2

. The larger of these two solutions is

2.5

.

Step-by-Step Solution

1

Multiply the entire equation by the least common denominator, which is

x(x-1)

, for

x \neq 0

and

x \neq 1

.

10(x-1) - 3x = 2x(x-1)

This clears the fractions from the rational equation.

2

Expand both sides of the equation.

10x - 10 - 3x = 2x^2 - 2x

Expanding the terms allows us to combine like terms.

3

Combine like terms on the left side and move all terms to one side to set the quadratic equation equal to zero.

2x^2 - 9x + 10 = 0

Setting the quadratic equation to zero is required to solve it by factoring.

4

Factor the quadratic equation by grouping.

(2x-5)(x-2) = 0

Factoring allows us to find the roots of the quadratic equation.

5

Solve for

x

by setting each factor equal to zero.

x = 2.5

or

x = 2

By the zero product property, at least one of the factors must be zero.

6

Check for extraneous solutions and select the larger real value.

Both

2

and

2.5

are valid because they do not make the original denominators zero. The larger value is

2.5

.

The question specifically asks for the larger of the two real solutions.

Key Concept

Solving rational equations by clearing denominators

Estimated Time:1m 30s

Question 123Question →

Let the function $h$ be defined by $h(x) = 3x - 5$ , and let the function $g$ be defined by $g(x) = x^2 + c$ for some constant $c$ . If $h(g(2)) = 10$ , what is the value of $c$ ?

Show answer & explanation

Answer: 1

Answer

The value of the constant

c

is

1

.

The correct answer is

1

. Evaluating the inner function

g(2)

gives

2^2 + c = 4 + c

. Applying the outer function

h(x)

to this expression gives

h(4 + c) = 3(4 + c) - 5

. Simplifying this expression results in

12 + 3c - 5 = 7 + 3c

. We are given that this value equals

10

, so

7 + 3c = 10

. Subtracting 7 from both sides gives

3c = 3

, which yields

c = 1

.

Step-by-Step Solution

1

Evaluate

g(2)

in terms of

c

g(2) = 4 + c

We substitute

x = 2

into the definition

g(x) = x^2 + c

to get

2^2 + c = 4 + c

.

2

Substitute

g(2)

into the definition of

h(x)

to express

h(g(2))

h(g(2)) = 3(4 + c) - 5

Since the composition is

h(g(2))

, we evaluate the function

h

at the input value

g(2) = 4 + c

.

3

Set

h(g(2)) = 10

and solve the linear equation for

c

c = 1

Expanding and simplifying the equation

3(4 + c) - 5 = 10

gives

12 + 3c - 5 = 10

, which reduces to

7 + 3c = 10

. Subtracting 7 gives

3c = 3

, so

c = 1

.

Key Concept

Function Composition and Parameter Evaluation

Estimated Time:1m 15s

Question 124Question →

A sporting goods company sells soccer balls and basketballs at two locations, North and South. The matrix $A$ represents the inventory of these balls at each location:

A = \begin{bmatrix} 50 & 30 \\ 40 & 60 \end{bmatrix}

where the rows represent the locations (Row 1: North, Row 2: South) and the columns represent the ball types (Column 1: soccer balls, Column 2: basketballs).

The matrix $B$ represents the price, in dollars, of each type of ball under two different pricing structures (Column 1: regular price, Column 2: sale price):

B = \begin{bmatrix} 20 & 15 \\ 25 & 18 \end{bmatrix}

where Row 1 represents soccer balls and Row 2 represents basketballs.

Which of the following matrices represents the total revenue, in dollars, the company would collect from selling its entire inventory at each location under both pricing structures?

\begin{bmatrix} 1,000 & 450 \\ 1,000 & 1,080 \end{bmatrix}

\begin{bmatrix} 1,600 & 1,500 \\ 1,970 & 1,830 \end{bmatrix}

\begin{bmatrix} 250 & 210 \\ 700 & 480 \end{bmatrix}

\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

\begin{bmatrix} 1,750 & 1,380 \\ 2,300 & 1,720 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}$

Answer

\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

To find the total revenue under both pricing structures at each location, we perform the matrix multiplication

AB

. The entry in row

i

, column

j

of the resulting matrix corresponds to the revenue generated at location

i

under pricing structure

j

. Applying the row-by-column multiplication rule gives the correct matrix containing

1,750

,

1,290

,

2,300

, and

1,680

.

Step-by-Step Solution

1

Set up the matrix multiplication to compute the total revenue.

The total revenue matrix is represented by the product

AB

, where we multiply the

2 \times 2

inventory matrix

A

by the

2 \times 2

price matrix

B

.

Multiplying the quantities of items by their respective unit prices yields the total revenue.

2

Calculate the entries for the first row of the product matrix, representing the North location.

Row 1, Column 1 (North, Regular):

50(20) + 30(25) = 1,000 + 750 = 1,750

.
Row 1, Column 2 (North, Sale):

50(15) + 30(18) = 750 + 540 = 1,290

.

The first row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the North location.

3

Calculate the entries for the second row of the product matrix, representing the South location.

Row 2, Column 1 (South, Regular):

40(20) + 60(25) = 800 + 1,500 = 2,300

.
Row 2, Column 2 (South, Sale):

40(15) + 60(18) = 600 + 1,080 = 1,680

.

The second row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the South location.

4

Construct the final product matrix from the calculated values.

AB = \begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

Placing the computed row values in their respective positions yields the completed revenue matrix.

Key Concept

Matrix multiplication is used to combine quantity data and unit cost data systematically across multiple categories and scenarios.

Question 125Question →

A sporting goods chain has two locations. Location 1 sells 10 soccer balls and 15 basketballs. Location 2 sells 8 soccer balls and 20 basketballs. This is represented by the sales matrix $A = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix}$ . Soccer balls cost $15 each and basketballs cost$ 25 each, represented by the price matrix $B = \begin{bmatrix} 15 \\ 25 \\end{bmatrix}$ . What is the product $AB$ , which represents the total revenue generated by each location?

\begin{bmatrix} 150 \\ 500 \\end{bmatrix}

\begin{bmatrix} 350 \\ 725 \\end{bmatrix}

\begin{bmatrix} 525 \\ 520 \\end{bmatrix}

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

\begin{bmatrix} 525 & 620 \\end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 525 \\ 620 \\end{bmatrix}$

Answer

The column matrix with elements 525 in the first row and 620 in the second row

The correct answer is the column matrix containing 525 and 620. This is found by multiplying each row of the sales matrix (representing each location) by the column of the price matrix (representing prices of the items):

10(15) + 15(25) = 525

for the first row, and

8(15) + 20(25) = 620

for the second row.

Step-by-Step Solution

1

Set up the matrix multiplication of the

2 \times 2

sales matrix

A

and the

2 \times 1

price matrix

B

.

AB = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix} \begin{bmatrix} 15 \\ 25 \\end{bmatrix}

To find the total revenue for each location, we multiply the sales of each item by its corresponding price.

2

Multiply the first row of

A

by the column of

B

to find the element in the first row of the product matrix (revenue for Location 1).

10(15) + 15(25) = 150 + 375 = 525

This calculates the total revenue for Location 1 by summing the revenue from soccer balls (

10 \times 15

) and basketballs (

15 \times 25

).

3

Multiply the second row of

A

by the column of

B

to find the element in the second row of the product matrix (revenue for Location 2).

8(15) + 20(25) = 120 + 500 = 620

This calculates the total revenue for Location 2 by summing the revenue from soccer balls (

8 \times 15

) and basketballs (

20 \times 25

).

4

Combine the results into a

2 \times 1

resulting matrix.

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

Since matrix

A

has dimensions

2 \times 2

and matrix

B

has dimensions

2 \times 1

, their product

AB

must have dimensions

2 \times 1

.

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix, matching corresponding elements and adding the products.

Alternative Method

Instead of matrix multiplication, the revenue can be computed using standard arithmetic: Location 1 revenue is

10 \times \$15 + 15 \times \$25 = \$525

and Location 2 revenue is

8 \times \$15 + 20 \times \$25 = \$620

. The resulting column matrix is

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

.

Estimated Time:1m 0s

Question 126Question →

The first term of a geometric sequence is $3$ , and the second term is $6$ . What is the $5$ th term of this sequence?

15

24

48

96

1296

Show answer & explanation

Answer: $48$

Answer

The fifth term of the sequence is

48

.

The first term of the geometric sequence is

a_1 = 3

, and the second term is

a_2 = 6

. The common ratio

r

is

6 \div 3 = 2

. The

n

th term of a geometric sequence is given by

a_n = a_1 \cdot r^{n-1}

. For the fifth term (

n=5

), the calculation is

3 \cdot 2^{5-1} = 3 \cdot 2^4 = 3 \cdot 16 = 48

. This matches the correct value.

Step-by-Step Solution

1

Find the common ratio (

r

) of the geometric sequence.

r = \frac{6}{3} = 2

The common ratio of a geometric sequence is found by dividing any term by the preceding term.

2

Apply the formula for the

n

th term of a geometric sequence,

a_n = a_1 \cdot r^{n-1}

, to find the fifth term.

a_5 = 3 \cdot 2^{5-1} = 3 \cdot 2^4

We substitute

a_1 = 3

,

r = 2

, and

n = 5

into the standard formula.

3

Evaluate the expression.

a_5 = 3 \cdot 16 = 48

Calculate

2^4 = 16

first, then multiply by

3

according to the order of operations.

Key Concept

Finding a specific term in a geometric sequence using the general formula

Question 127Question →

The matrices $M$ and $N$ are defined as follows:

M = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix}, \quad N = \begin{bmatrix} 2 & 5 \\ 0 & -1 \end{bmatrix}

If the product matrix $P$ is defined by the equation $P = MN$ , what is the value of the element in the first row and second column of $P$ ?

Show answer & explanation

Answer: 22

Answer

The value of the element in the first row and second column of

P

is 22.

To find the element in the first row and second column of the product matrix

P = MN

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

M

(

[4, -2]

) and the second column of matrix

N

(

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

):

4(5) + (-2)(-1) = 20 + 2 = 22

.

Step-by-Step Solution

1

Determine the row of the left matrix and the column of the right matrix needed for the target element.

To find the element in row 1, column 2 of

P = MN

, use the first row of

M

, which is

[4, -2]

, and the second column of

N

, which is

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

.

The entry in row

i

and column

j

of a product matrix is the dot product of row

i

of the first matrix and column

j

of the second matrix.

2

Compute the dot product of the selected row and column.

(4 \times 5) + (-2 \times -1) = 20 + 2

Multiply corresponding elements and sum the products.

3

Simplify to get the final numeric answer.

22

Adding 20 and 2 yields 22.

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix.

Question 128Question →

One of the solutions to the quadratic equation $0.5x^2 - x + c = 0$ is $1 + i$ , where $i = \sqrt{-1}$ and $c$ is a constant. What is the value of $c$ ?

0

1 - 2i

1

-1

1 + i

Show answer & explanation

Answer: 1

Answer

The constant

c

is equal to

1

.

Substituting the given solution

x = 1 + i

into the equation

0.5x^2 - x + c = 0

gives

0.5(1+i)^2 - (1+i) + c = 0

. Since

(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i

, the equation simplifies to

0.5(2i) - 1 - i + c = 0

, which is

i - 1 - i + c = 0

. Simplifying this yields

-1 + c = 0

, so the value of

c

is

1

.

Step-by-Step Solution

1

Substitute the solution

x = 1 + i

into the quadratic equation

0.5x^2 - x + c = 0

.

0.5(1+i)^2 - (1+i) + c = 0

Since

1 + i

is a solution to the equation, substituting it for

x

must satisfy the equation.

2

Expand the squared term

(1+i)^2

using the binomial square formula and the identity

i^2 = -1

.

(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i

Expanding

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

with

a=1

and

b=i

, and replacing

i^2

with

-1

simplifies the term.

3

Substitute

2i

back into the equation and simplify.

0.5(2i) - (1+i) + c = 0 \Rightarrow i - 1 - i + c = 0 \Rightarrow -1 + c = 0

Multiplying

0.5

by

2i

yields

i

, distributing the negative sign across

(1+i)

yields

-1 - i

, and combining like terms leaves

-1 + c = 0

.

4

Solve the linear equation for

c

.

c = 1

Adding

1

to both sides isolates the variable

c

.

Key Concept

Solving quadratic equations with complex roots by substitution and algebraic manipulation.

Question 129Question →

For real numbers $x$ where the functions are defined, let $f(x) = \frac{x}{x^2 - 4}$ and $g(x) = \frac{x+3}{x-1}$ . What is the set of all real numbers $x$ for which the composite function $h(x) = f(g(x))$ is undefined?

\left\{1\right\}

\left\{-2, 1, 2\right\}

\left\{1, 5\right\}

\left\{-\frac{1}{3}, 1, 5\right\}

\left\{-\frac{1}{3}, -2, 1, 2, 5\right\}

Show answer & explanation

Answer: $\left\{-\frac{1}{3}, 1, 5\right\}$

Answer

The set containing negative one-third, one, and five

The correct answer is the set containing negative one-third, one, and five. The composite function

f(g(x))

is undefined when the inner function

g(x)

is undefined (which occurs at

x = 1

) or when the output of

g(x)

equals a value that makes the outer function

f(x)

undefined. The outer function

f(x)

is undefined when

x^2 - 4 = 0

, which gives

x = \pm 2

. Setting

g(x) = 2

leads to

x = 5

, and setting

g(x) = -2

leads to

x = -\frac{1}{3}

. Collecting all these values gives the complete set of restrictions.

Step-by-Step Solution

1

Determine the domain restriction of the inner function

x \neq 1

For the composite function

h(x) = f(g(x))

to be defined, the input

x

must first be in the domain of the inner function

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

. Thus, the denominator cannot be zero, which gives

x - 1 \neq 0 \implies x \neq 1

.

2

Determine the restricted inputs for the outer function

u \neq 2

and

u \neq -2

The outer function

f(u) = \frac{u}{u^2 - 4}

is undefined when its denominator is zero. Solving

u^2 - 4 = 0

yields

u^2 = 4 \implies u = 2

or

u = -2

. Therefore, the output of the inner function,

g(x)

, cannot equal

2

or

-2

.

3

Solve the equation for the first restricted value of the outer function

x = 5

Setting the inner function equal to

2

gives

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

. Multiplying both sides by

x-1

results in

x+3 = 2(x-1) \implies x+3 = 2x-2 \implies x = 5

.

4

Solve the equation for the second restricted value of the outer function

x = -\frac{1}{3}

Setting the inner function equal to

-2

gives

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

. Multiplying both sides by

x-1

results in

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

.

5

Combine all restricted values of

x

S = \left\{-\frac{1}{3}, 1, 5\right\}

The values of

x

for which the composite function is undefined are

x = 1

(from the inner function's domain restriction) along with

x = 5

and

x = -\frac{1}{3}

(which make the inner function's output equal to the outer function's restricted inputs).

Key Concept

Domain of Composite Functions

Estimated Time:2m 30s

Question 130Question →

A catering service is organizing two events: a wedding and a corporate gala. The matrix $P$ represents the number of Deluxe packages ( $d$ ) and Premium packages ( $p$ ) ordered for each event:

P = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix}

where the rows represent the Wedding and the Gala respectively, and the columns represent Deluxe and Premium packages respectively.

The matrix $R$ represents the requirements (in units) of protein and vegetables for each package type:

R = \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix}

where the rows represent Deluxe and Premium packages respectively, and the columns represent protein and vegetables respectively.

The total units of ingredients required for each event are represented by the product matrix $T = P \times R$ .

If the total vegetables required for the wedding is 220 units, and the total protein required for the corporate gala is 560 units, what is the value of the sum $x + y$ ?

11

54.5

82.8

83

145.5

Show answer & explanation

Answer: 83

Answer

83

The correct option is the value 83. Setting up the product matrix

T = P \times R

, the entry in row 1, column 2 (representing wedding vegetables) is

50(2) + 40y = 220

, which simplifies to

y = 3

. The entry in row 2, column 1 (representing gala protein) is

80(3) + 4x = 560

, which simplifies to

x = 80

. Adding these values together gives

x + y = 83

.

Step-by-Step Solution

1

Set up the matrix product expression

T = P \times R = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix} \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix} = \begin{bmatrix} 50(3) + 40(4) & 50(2) + 40(y) \\ 80(3) + 4x & 80(2) + xy \end{bmatrix}

We multiply row elements of the first matrix by column elements of the second matrix to obtain the entries of the product matrix.

2

Isolate the equation for vegetables required for the wedding

50(2) + 40y = 220 \implies 100 + 40y = 220 \implies 40y = 120 \implies y = 3

The wedding corresponds to the first row of matrix

P

, and vegetables correspond to the second column of matrix

R

. Their dot product equals the total vegetables for the wedding.

3

Isolate the equation for protein required for the corporate gala

80(3) + 4x = 560 \implies 240 + 4x = 560 \implies 4x = 320 \implies x = 80

The corporate gala corresponds to the second row of matrix

P

, and protein corresponds to the first column of matrix

R

. Their dot product equals the total protein for the gala.

4

Find the sum of the variables

x + y = 80 + 3 = 83

We add the solved values of

x

and

y

to obtain the requested sum.

Key Concept

Matrix Multiplication and Word Problem Translation

Estimated Time:3m 0s

Question 131Question →

If $3 - 2i$ is a root of the quadratic equation $x^2 + bx + c = 0$ , where $b$ and $c$ are real numbers and $i = \sqrt{-1}$ , what is the value of $b + c$ ?

-11

-5

-1

7

19

Show answer & explanation

Answer: 7

Answer

7

The correct answer is

7

. Since the quadratic equation has real coefficients, the roots must be complex conjugates. The conjugate of the root

3 - 2i

is

3 + 2i

. Vieta's formulas show that the sum of the roots is

-b

, meaning

(3 - 2i) + (3 + 2i) = 6 = -b

, which yields

b = -6

. The product of the roots is

c

, meaning

(3 - 2i)(3 + 2i) = 9 - 4i^2 = 9 - 4(-1) = 13

. Adding these coefficients together gives

b + c = -6 + 13 = 7

.

Step-by-Step Solution

1

Identify the second root of the quadratic equation

The second root is

3 + 2i

.

Since the quadratic equation has real coefficients, the Complex Conjugate Theorem dictates that if a complex number is a root, its complex conjugate must also be a root.

2

Determine the coefficient

b

using the sum of the roots

b = -6

According to Vieta's formulas, the sum of the roots of the equation

x^2 + bx + c = 0

is equal to

-b

. Thus,

(3 - 2i) + (3 + 2i) = 6 = -b

, which simplifies to

b = -6

.

3

Determine the constant term

c

using the product of the roots

c = 13

According to Vieta's formulas, the product of the roots of the equation

x^2 + bx + c = 0

is equal to

c

. Thus,

(3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4i^2

. Substituting

i^2 = -1

yields

9 - 4(-1) = 9 + 4 = 13

.

4

Calculate the value of

b + c

b + c = 7

Substitute the calculated values of

b

and

c

into the expression:

-6 + 13 = 7

.

Key Concept

The Complex Conjugate Theorem states that complex roots of polynomials with real coefficients occur in conjugate pairs. Vieta's formulas state that for a quadratic equation

x^2 + bx + c = 0

, the sum of the roots is

-b

and the product of the roots is

c

.

Question 132Question →

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 133Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 = 17$ and a line is defined by the equation $y = x - 3$ . If $(x, y)$ represents the intersection point of the circle and the line that lies in the first quadrant, what is the value of $x + y$ ?

1

3

5

7

-5

Show answer & explanation

Answer: 5

Answer

The sum of the coordinates of the first-quadrant intersection point is 5.

The correct answer is the sum of the coordinates of the first-quadrant intersection point. By substituting

y = x - 3

into the circle's equation, we get

x^2 + (x - 3)^2 = 17

. Expanding the squared term gives

x^2 + x^2 - 6x + 9 = 17

, which simplifies to

2x^2 - 6x - 8 = 0

. Dividing the entire equation by 2 yields

x^2 - 3x - 4 = 0

. Factoring this quadratic equation gives

(x - 4)(x + 1) = 0

, which has solutions

x = 4

and

x = -1

. Because the intersection point must lie in the first quadrant, the

x

-coordinate must be positive, so we choose

x = 4

. Substituting

x = 4

back into the linear equation gives

y = 4 - 3 = 1

. The sum of these coordinates is

4 + 1 = 5

.

Step-by-Step Solution

1

Substitute the expression for

y

from the linear equation into the circle's equation.

x^2 + (x - 3)^2 = 17

This eliminates

y

so that we can solve for

x

.

2

Expand the squared binomial and combine like terms to write the equation in standard quadratic form.

2x^2 - 6x - 8 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

, and adding

x^2

and subtracting 17 from both sides gives the quadratic equation.

3

Simplify the quadratic equation by dividing all terms by 2, then factor the resulting quadratic expression.

(x - 4)(x + 1) = 0

Dividing by 2 gives

x^2 - 3x - 4 = 0

, which factors into

(x - 4)(x + 1) = 0

since

(-4) \times 1 = -4

and

-4 + 1 = -3

.

4

Solve for

x

and select the positive solution since the point lies in the first quadrant.

x = 4

The solutions are

x = 4

and

x = -1

. In the first quadrant, both coordinates must be positive, so we choose

x = 4

.

5

Substitute

x = 4

back into the linear equation to find the corresponding

y

-coordinate.

y = 1

Using

y = x - 3

with

x = 4

gives

y = 4 - 3 = 1

.

6

Calculate the sum of the coordinates

x + y

.

4 + 1 = 5

The question asks for the value of

x + y

.

Key Concept

Solving a system consisting of a linear equation and a quadratic circle equation by substitution, factoring the resulting quadratic equation, and applying quadrant constraints.

Question 134Question →

Two functions, $f$ and $g$ , are defined as $f(x) = (x - 4)^2$ and $g(x) = |2x - 3|$ . What is the value of the composite function $f(g(x))$ evaluated at $x = -1$ ?

1

9

47

81

125

Show answer & explanation

Answer: 1

Answer

1

To find

f(g(-1))

, we first evaluate the inner function

g

at

x = -1

. Substituting

-1

into

g(x) = |2x - 3|

gives

g(-1) = |2(-1) - 3| = |-5| = 5

. Next, we use this result as the input for the outer function

f

, evaluating

f(5)

. Substituting

5

into

f(x) = (x - 4)^2

gives

f(5) = (5 - 4)^2 = 1^2 = 1

. This corresponds to the option with value 1.

Step-by-Step Solution

1

Evaluate the inner function

g(-1)

first.

g(-1) = |2(-1) - 3| = |-2 - 3| = |-5| = 5

The input to the outer function of a composition is the output of the inner function.

2

Evaluate the outer function

f(x)

at the result from the previous step.

f(5) = (5 - 4)^2 = 1^2 = 1

We substitute the value of

g(-1)

, which is

5

, into the function

f(x)

to find the final value of the composition.

Key Concept

Evaluating a composite function involves applying the inner function first, then using that output as the input for the outer function.

Question 135Question →

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 136Question →

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 137Question →

An entrepreneur starts a company with an operating budget of $100,000$ in its first year. For each of the next 4 years (years 2 through 5), the budget increases by a constant amount of $d$ dollars each year. For years 6 through 8, the budget increases geometrically, where the budget in year 6 is $1.5$ times the budget in year 5, and the budget increases by $50\%$ each year thereafter. If the total operating budget over the first 8 years is $1,597,500$ dollars, what is the value of $d$ ?

4,300

10,000

17,500

33,900

38,500

Show answer & explanation

Answer: $10,000$

Answer

The common difference is

10,000

.

The correct answer is found by setting up the sum of the first 5 years of the arithmetic sequence and the subsequent 3 years of the geometric sequence. Summing the expressions for all 8 years gives

1,212,500 + 38.5d

. Equating this expression to the total budget of

1,597,500

and solving for

d

yields

10,000

.

Step-by-Step Solution

1

Express the budget for the first 5 years as an arithmetic sequence and find their sum.

S_5 = 100,000 + (100,000 + d) + (100,000 + 2d) + (100,000 + 3d) + (100,000 + 4d) = 500,000 + 10d

The budget starts at

100,000

in year 1 and increases by a constant amount

d

each year through year 5.

2

Express the budgets for years 6 through 8 as a geometric sequence starting from

1.5

times the year 5 budget.

Year 6:

1.5(100,000 + 4d) = 150,000 + 6d

; Year 7:

1.5(150,000 + 6d) = 225,000 + 9d

; Year 8:

1.5(225,000 + 9d) = 337,500 + 13.5d

. The sum of these 3 years is

712,500 + 28.5d

.

The budget increases by a factor of

1.5

(or

50\%

) each year starting from year 6.

3

Combine the sums of both sequences to represent the total 8-year budget and solve for

d

.

Total =

(500,000 + 10d) + (712,500 + 28.5d) = 1,212,500 + 38.5d

. Setting this equal to the given total:

1,212,500 + 38.5d = 1,597,500 \implies 38.5d = 385,000 \implies d = 10,000

.

The total budget over the 8 years is the sum of the budgets of the individual years.

Key Concept

Combining arithmetic and geometric sequences in multi-step word problems.

Alternative Method

Instead of calculating each geometric year sequentially, the sum of the geometric sequence for years 6 to 8 can be calculated using the geometric series sum formula

S_n = a_1 \frac{1 - r^n}{1 - r}

with

a_1 = 1.5(100,000 + 4d)

and

r = 1.5

over

n = 3

terms.

Estimated Time:2m 30s

Question 138Question →

A local farm sells organic fruit baskets at a weekend market. The number of small and large baskets sold on Saturday and Sunday is represented by matrix $Q$ , where row 1 represents Saturday, row 2 represents Sunday, column 1 represents small baskets, and column 2 represents large baskets:

Q = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix}

The selling price and the production cost, in dollars, for each type of basket are represented by matrix $P$ , where row 1 represents small baskets, row 2 represents large baskets, column 1 represents the selling price, and column 2 represents the production cost:

P = \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

The product matrix $R = QP$ represents the total revenue and total production cost for each day. Which of the following matrices represents $R$ ?

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

\begin{bmatrix} 600 & 240 \\ 500 & 600 \end{bmatrix}

\begin{bmatrix} 760 & 850 \\ 1240 & 1350 \end{bmatrix}

\begin{bmatrix} 1350 & 1550 \\ 680 & 760 \end{bmatrix}

\begin{bmatrix} 1450 & 680 \\ 1550 & 660 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}$

Answer

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

The correct product matrix is obtained by performing matrix multiplication

QP

. Multiplying the first row of

Q

by the first column of

P

yields the Saturday revenue:

40 \times 15 + 30 \times 25 = 1350

. Multiplying the first row of

Q

by the second column of

P

yields the Saturday cost:

40 \times 8 + 30 \times 12 = 680

. Repeating this for the second row of

Q

(Sunday) yields the Sunday revenue of

1550

and Sunday cost of

760

. Thus, the resulting matrix is the one with row 1 equal to

[1350, 680]

and row 2 equal to

[1550, 760]

.

Step-by-Step Solution

1

Set up the matrix multiplication product

R = QP

.

R = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix} \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

To find the total revenue and production cost for Saturday and Sunday, we must multiply the quantity matrix by the price-cost matrix.

2

Calculate the elements of the first row of the product matrix

R

, representing Saturday's revenue and cost.

Row 1, Column 1 (Saturday Revenue):

40(15) + 30(25) = 600 + 750 = 1350

. Row 1, Column 2 (Saturday Cost):

40(8) + 30(12) = 320 + 360 = 680

.

Multiply the first row of matrix

Q

by the columns of matrix

P

to determine Saturday's financial values.

3

Calculate the elements of the second row of the product matrix

R

, representing Sunday's revenue and cost.

Row 2, Column 1 (Sunday Revenue):

20(15) + 50(25) = 300 + 1250 = 1550

. Row 2, Column 2 (Sunday Cost):

20(8) + 50(12) = 160 + 600 = 760

.

Multiply the second row of matrix

Q

by the columns of matrix

P

to determine Sunday's financial values.

4

Combine the calculated elements into the final

2 \times 2

matrix.

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

Placing the values in their corresponding row and column positions yields the complete product matrix.

Key Concept

Matrix Multiplication in Applied Word Problems

Estimated Time:1m 30s

Question 139Question →

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 140Question →

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.

Intermediate Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Answer

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Answer

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Common Mistakes

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Common Mistakes

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Answer

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Key Concept

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