All practice questions

Question 161Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has a slope of $\frac{2}{3}$ and passes through the point $(6, 5)$ . If $a + b = 5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

15

Rearranging the linear equation

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows the slope is

-\frac{a}{b}

. Setting

-\frac{a}{b} = \frac{2}{3}

gives

2b = -3a

, or

b = -1.5a

. Substituting this into

a + b = 5

yields

a - 1.5a = 5

, which simplifies to

-0.5a = 5

, so

a = -10

. This means

b = -1.5(-10) = 15

. Substituting the values of

a = -10

and

b = 15

along with the point

(6, 5)

into the equation

ax + by = c

yields

(-10)(6) + (15)(5) = 15

, so the value of

c

is

15

.

Step-by-Step Solution

1

Rewrite the standard form equation

ax + by = c

in slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

This allows the identification of the slope of the line in terms of the coefficients

a

and

b

.

2

Set the slope expression equal to the given slope of

\frac{2}{3}

and solve for

b

in terms of

a

.

b = -1.5a

The slope of the line is

-\frac{a}{b}

, so

-\frac{a}{b} = \frac{2}{3} \implies 2b = -3a \implies b = -1.5a

.

3

Substitute

b = -1.5a

into the given equation

a + b = 5

and solve for

a

, then find

b

.

a = -10

and

b = 15

Substituting gives

a - 1.5a = 5 \implies -0.5a = 5 \implies a = -10

. Substituting

a = -10

back into the relationship gives

b = -1.5(-10) = 15

.

4

Substitute

a = -10

,

b = 15

, and the point

(6, 5)

into the equation

ax + by = c

and solve for

c

.

c = 15

(-10)(6) + (15)(5) = c \implies -60 + 75 = 15 \implies c = 15

.

Key Concept

Linear Equations in Two Variables

Question 162Question →

If the solution to the inequality $a(2 - 3x) - 4(x + 3) \ge 12$ , where $a$ is a constant, is $x \le -1$ , what is the value of $a$ ?

Show answer & explanation

Answer: 4

Answer

The value of

a

is

4

.

To solve the inequality

a(2 - 3x) - 4(x + 3) \ge 12

, we first expand it to get

2a - 3ax - 4x - 12 \ge 12

. Grouping the

x

terms gives

(-3a - 4)x \ge 24 - 2a

. Since the inequality's solution is

x \le -1

, the direction of the inequality must flip, which means the coefficient of

x

, namely

-3a - 4

, must be negative. Dividing both sides by this coefficient gives the boundary value of the inequality as

\frac{24 - 2a}{-3a - 4}

. Setting this boundary equal to

-1

yields

24 - 2a = 3a + 4

, which simplifies to

5a = 20

, or

a = 4

. Since

a = 4

makes the coefficient

-3(4) - 4 = -16

negative, the solution holds.

Step-by-Step Solution

1

Expand the inequality using the distributive property.

2a - 3ax - 4x - 12 \ge 12

This allows us to separate and group the terms containing the variable

x

and the constant terms.

2

Group like terms and isolate the variable terms on the left-hand side.

(-3a - 4)x \ge 24 - 2a

By combining the coefficients of

x

and adding

12 - 2a

to both sides, we prepare the inequality to solve for

x

.

3

Determine the effect of dividing by the variable's coefficient.

Since the given solution is

x \le -1

, the inequality sign flipped from

\ge

to

\le

. Therefore, the coefficient

-3a - 4

must be negative.

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

4

Set the boundary value of the solution equal to

-1

and solve for

a

.

a = 4

Setting the boundary of the inequality

\frac{24 - 2a}{-3a - 4}

equal to

-1

allows us to find the specific constant

a

that produces this solution set.

Key Concept

Solving linear inequalities in one variable involving parameters and sign flips.

Question 163Question →

The graphs of the equations $y = x^2 - 10$ and $y = 2x - 2$ intersect at the point $(x, y)$ in the first quadrant. What is the value of $y$ ?

Show answer & explanation

Answer: 6

Answer

The value of

y

is 6.

Equating the equations gives

x^2 - 10 = 2x - 2

. Moving all terms to one side yields

x^2 - 2x - 8 = 0

, which factors as

(x - 4)(x + 2) = 0

. Since the point is in the first quadrant, both coordinates must be positive, so we use

x = 4

. Substituting

x = 4

into the linear equation gives

y = 2(4) - 2 = 6

.

Step-by-Step Solution

1

Equate the two expressions for

y

x^2 - 10 = 2x - 2

Since both equations define

y

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

2

Rewrite the equation in standard quadratic form

ax^2 + bx + c = 0

x^2 - 2x - 8 = 0

Subtract

2x

and add

2

to both sides of the equation to set it equal to zero.

3

Factor the quadratic equation

(x - 4)(x + 2) = 0

Find two integers that multiply to

-8

and add to

-2

, which are

-4

and

2

.

4

Solve for the possible values of

x

x = 4

or

x = -2

Set each factor equal to zero and solve.

5

Determine the positive

x

-coordinate and find

y

y = 6

For the point to be in the first quadrant, both coordinates must be positive. Thus, we select

x = 4

and substitute it into the linear equation:

y = 2(4) - 2 = 6

.

Key Concept

Solving a system of linear and quadratic equations via substitution.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

x^2 - 2x - 8 = 0

:

x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2}

. This yields

x = 4

and

x = -2

. Then substitute the positive root to find

y

.

Estimated Time:1m 30s

Question 164Question →

If the expression $\frac{3x^2 + 10x - 8}{x + k}$ is equivalent to $3x - 2$ for all $x \neq -k$ , where $k$ is a positive constant, what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the constant

k

is 4.

For the expressions to be equivalent for all values of

x

, the numerator

3x^2 + 10x - 8

must be equal to the product of the denominator

x + k

and the simplified quotient

3x - 2

. Expanding the product yields

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

. Equating the constant terms on both sides gives

-8 = -2k

, which results in

k = 4

. Alternatively, equating the coefficients of the linear terms gives

10 = 3k - 2

, which also yields

k = 4

.

Step-by-Step Solution

1

Multiply both sides of the equivalence by the denominator

x + k

.

3x^2 + 10x - 8 = (3x - 2)(x + k)

To eliminate the fraction and set up a polynomial identity.

2

Expand the right side of the equation.

3x^2 + 10x - 8 = 3x^2 + (3k - 2)x - 2k

To express the right side in standard quadratic form for coefficient comparison.

3

Equate the constant terms to solve for

k

.

-8 = -2k \implies k = 4

Since the expressions are equivalent for all values of

x

, their corresponding coefficients and constants must be equal.

Key Concept

Equating coefficients of equivalent polynomial expressions

Question 165Question →

If the expression $\frac{4x^3 - 2x^2 + 7x + 7}{2x^2 + 1}$ is equivalent to $ax + b + \frac{cx + d}{2x^2 + 1}$ for all values of $x$ , where $a$ , $b$ , $c$ , and $d$ are constants, what is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 14

Answer

The value of

a + b + c + d

is 14.

Performing polynomial long division on

\frac{4x^3 - 2x^2 + 7x + 7}{2x^2 + 1}

yields a quotient of

2x - 1

and a remainder of

5x + 8

. Matching this to the form

ax + b + \frac{cx + d}{2x^2 + 1}

gives

a = 2

,

b = -1

,

c = 5

, and

d = 8

. Summing these values gives

2 + (-1) + 5 + 8 = 14

.

Step-by-Step Solution

1

Divide the leading term of the numerator by the leading term of the denominator to find the first term of the quotient.

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

. Multiplying

2x(2x^2 + 1) = 4x^3 + 2x

. Subtracting this from the numerator yields

-2x^2 + 5x + 7

.

To initiate the polynomial division process.

2

Divide the leading term of the remaining polynomial by the leading term of the divisor to find the second term of the quotient.

\frac{-2x^2}{2x^2} = -1

. Multiplying

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

. Subtracting this from the remaining polynomial yields

5x + 8

.

To find the next term of the quotient and determine the remainder.

3

Write the expression in the quotient-remainder form and identify the values of the constants

a

,

b

,

c

, and

d

.

The expression is equivalent to

2x - 1 + \frac{5x + 8}{2x^2 + 1}

, so

a = 2

,

b = -1

,

c = 5

, and

d = 8

.

To match the given algebraic form of the expression.

4

Calculate the sum of the constants

a + b + c + d

.

2 + (-1) + 5 + 8 = 14

.

To find the final value requested by the question.

Key Concept

Equivalent Algebraic Expressions

Question 166Question →

For all $x > 8$ , the expression $\frac{x^{8/3} - 8x^{5/3}}{x^{4/3} - 4x^{2/3}} \cdot \frac{x^{1/3} + 2}{x^{4/3} + 2x + 4x^{2/3}}$ is equivalent to $x^a$ , where $a$ is a constant. What is the value of $\frac{1}{a}$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

Factoring the numerator and denominator of the first fraction yields

\frac{x^{5/3}(x^{1/3}-2)(x^{2/3}+2x^{1/3}+4)}{x^{2/3}(x^{1/3}-2)(x^{1/3}+2)}

, which simplifies to

\frac{x(x^{2/3}+2x^{1/3}+4)}{x^{1/3}+2}

. Factoring the denominator of the second fraction yields

\frac{x^{1/3}+2}{x^{2/3}(x^{2/3}+2x^{1/3}+4)}

. Multiplying these two simplified expressions cancels the common terms

(x^{1/3}+2)

and

(x^{2/3}+2x^{1/3}+4)

, leaving

\frac{x}{x^{2/3}} = x^{1/3}

. Therefore,

a = \frac{1}{3}

, and the value of the reciprocal

\frac{1}{a}

is

3

.

Step-by-Step Solution

1

Factor the numerator and the denominator of the first fraction.

The first fraction becomes

\frac{x(x^{2/3} + 2x^{1/3} + 4)}{x^{1/3} + 2}

.

Factoring out

x^{5/3}

from the numerator gives

x^{5/3}(x-8)

, and factoring out

x^{2/3}

from the denominator gives

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

. Using the difference of cubes

x - 8 = (x^{1/3} - 2)(x^{2/3} + 2x^{1/3} + 4)

and the difference of squares

x^{2/3} - 4 = (x^{1/3} - 2)(x^{1/3} + 2)

, we can cancel the common factor

(x^{1/3} - 2)

.

2

Factor the denominator of the second fraction.

The second fraction becomes

\frac{x^{1/3} + 2}{x^{2/3}(x^{2/3} + 2x^{1/3} + 4)}

.

Factoring out

x^{2/3}

from the expression

x^{4/3} + 2x + 4x^{2/3}

reveals a common quadratic-like term

(x^{2/3} + 2x^{1/3} + 4)

that can be used for cancellation.

3

Multiply the two rational expressions together and simplify.

x^{1/3}

Multiplying the simplified fractions allows us to cancel the common binomial term

(x^{1/3} + 2)

and the trinomial term

(x^{2/3} + 2x^{1/3} + 4)

, leaving

\frac{x}{x^{2/3}} = x^{1 - 2/3} = x^{1/3}

.

4

Find the value of

\frac{1}{a}

.

3

Since the expression is equivalent to

x^a

, we identify

a = \frac{1}{3}

. Taking the reciprocal of

a

gives

\frac{1}{1/3} = 3

.

Key Concept

Equivalent Algebraic Expressions

Question 167Question →

An electric vehicle's battery is being charged. The charge of the battery, $C$ , as a percentage of its full capacity, $t$ minutes after the charging begins is modeled by the equation $C = 1.2t + 18$ , where $t \le 60$ . According to the model, what was the battery's charge percentage when the charging began?

Show answer & explanation

Answer: 18

Answer

The battery's charge percentage when the charging began was 18.

The linear relationship is given by the equation

C = 1.2t + 18

. The constant term in this linear model, 18, represents the y-intercept, which is the value of

C

when

t = 0

. In the context of this scenario,

t = 0

represents the time when the charging began, and

C

represents the charge percentage. Substituting

t = 0

into the equation yields

C = 1.2(0) + 18 = 18

. Therefore, the battery's charge percentage was 18 when the charging began.

Step-by-Step Solution

1

Determine the value of the independent variable

t

when charging began.

t = 0

The initial state or the start of the charging process corresponds to a time of 0 minutes.

2

Substitute

t = 0

into the equation

C = 1.2t + 18

.

C = 18

Evaluating the linear equation at

t = 0

yields the constant term, representing the initial charge percentage.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 168Question →

The expression $(3x^2 + 5x) - (x^2 - kx)$ , where $k$ is a constant, can be rewritten as $2x^2 + 12x$ . What is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

To find the value of

k

, we first simplify the expression

(3x^2 + 5x) - (x^2 - kx)

by distributing the subtraction sign to both terms inside the second set of parentheses. This yields

3x^2 + 5x - x^2 + kx

. Next, we group and combine like terms to get

(3x^2 - x^2) + (5x + kx) = 2x^2 + (5+k)x

. Since this expression is equivalent to

2x^2 + 12x

for all values of

x

, the coefficients of corresponding terms must be equal. Equating the coefficients of

x

gives

5+k = 12

. Subtracting 5 from both sides yields

k = 7

.

Step-by-Step Solution

1

Distribute the negative sign to the terms in the second parentheses.

3x^2 + 5x - x^2 + kx

To remove the parentheses and simplify the expression.

2

Combine like terms.

2x^2 + (5 + k)x

Grouping the

x^2

terms and

x

terms simplifies comparison with the target expression.

3

Equate the coefficient of the

x

term to the corresponding coefficient in the target expression.

5 + k = 12 \implies k = 7

Equivalent expressions must have equal corresponding coefficients for all values of

x

.

Key Concept

Equivalence of polynomial expressions by combining like terms and equating coefficients

Question 169Question →

A customer opens a savings account with an initial deposit of $\$180$ and deposits $\$30$ at the end of each week. Two weeks later, a second customer opens a savings account with an initial deposit of $\$120$ and deposits $\$50$ at the end of each week. If neither customer makes any other deposits or withdrawals, after how many weeks from the time the first customer opened their account will both accounts have the same balance?

Show answer & explanation

Answer: 8

Answer

Both accounts will have the same balance after 8 weeks.

The correct answer is 8. Letting

w

represent the number of weeks since the first customer opened their account, the balance of the first account can be modeled by the expression

180 + 30w

. Since the second customer opens their account two weeks later, the number of weeks they have been depositing money is

w - 2

. Therefore, the balance of the second account can be modeled by the expression

120 + 50(w - 2)

. Setting the two expressions equal to find when the balances are the same gives the equation

180 + 30w = 120 + 50(w - 2)

. Distributing the

50

yields

180 + 30w = 120 + 50w - 100

. Simplifying the right side gives

180 + 30w = 50w + 20

. Subtracting

30w

and

20

from both sides of the equation yields

160 = 20w

. Dividing both sides by

20

gives

w = 8

.

Step-by-Step Solution

1

Set up expressions for the balance of each account after

w

weeks.

First account balance:

180 + 30w

; Second account balance:

120 + 50(w - 2)

The first customer has been depositing for

w

weeks. The second customer started 2 weeks later, so they have been depositing for

w - 2

weeks.

2

Set the two expressions equal to each other to find when the balances are equal.

180 + 30w = 120 + 50(w - 2)

We want to find the number of weeks

w

when the two account balances are equal.

3

Solve the linear equation for

w

.

w = 8

Distribute the 50 to get

180 + 30w = 120 + 50w - 100

. Combine constants to get

180 + 30w = 50w + 20

. Subtract

30w

and

20

from both sides to get

160 = 20w

. Divide by

20

to get

w = 8

.

Key Concept

Formulating and solving linear equations in one variable from real-world contexts

Question 170Question →

A delivery driver is loading a van with small boxes and large boxes. Each small box weighs $10$ pounds, and each large box weighs $35$ pounds. The total weight of the boxes in the van cannot exceed $1,000$ pounds. The driver must load at least $4$ times as many small boxes as large boxes. If the driver loads at least $10$ large boxes, what is the maximum number of large boxes that the driver can load?

Show answer & explanation

Answer: 13

Answer

13

To find the maximum number of large boxes,

y

, we set up the system of inequalities:

10x + 35y \le 1000

and

x \ge 4y

. To maximize

y

, we minimize

x

by setting

x = 4y

. Substituting this into the weight constraint gives

10(4y) + 35y \le 1000

, which simplifies to

75y \le 1000

, or

y \le 13.33

. The largest integer satisfying this inequality is 13, which also satisfies

y \ge 10

.

Step-by-Step Solution

1

Define variables for the quantities of each type of box.

Let

x

be the number of small boxes and

y

be the number of large boxes, where

x

and

y

are non-negative integers.

Establishing variables is necessary to translate the verbal constraints into algebraic inequalities.

2

Translate the given constraints into a system of linear inequalities.

The weight limit gives

10x + 35y \le 1000

. The requirement for at least 4 times as many small boxes as large boxes gives

x \ge 4y

. The minimum of 10 large boxes gives

y \ge 10

.

Representing the scenario mathematically allows us to find the boundaries of the solution set.

3

Express the weight inequality in terms of a single variable by using the boundary condition of the second inequality.

To maximize

y

, we want to minimize the weight contributed by the small boxes,

x

. The minimum value of

x

for any

y

is

x = 4y

. Substituting

x = 4y

into the weight inequality gives

10(4y) + 35y \le 1000

.

Finding the extreme case (minimum number of small boxes) provides the upper limit for the number of large boxes.

4

Simplify the inequality and solve for

y

.

40y + 35y \le 1000 \implies 75y \le 1000 \implies y \le \frac{1000}{75} \approx 13.33

.

This calculation determines the upper algebraic bound for the number of large boxes.

5

Determine the maximum integer value for

y

that satisfies the system.

Since the number of boxes must be an integer, the maximum integer value less than or equal to

13.33

is

13

. Since

13 \ge 10

, this satisfies all constraints.

Real-world quantities like boxes must be whole numbers, so we take the largest integer within the solution set.

Key Concept

Solving systems of linear inequalities in two variables to optimize a value under constraints.

Question 171Question →

A distributor plans to mix two coffee blends, Blend A and Blend B, to create a custom mixture. The table below shows the distribution of Colombian and Ethiopian coffee beans in each blend by weight:

Coffee Blend	Colombian Beans	Ethiopian Beans
Blend A	60%	40%
Blend B	20%	80%

The distributor wants the final custom mixture to contain exactly 14 kilograms of Colombian coffee beans and 16 kilograms of Ethiopian coffee beans. How many kilograms of Blend A should the distributor use to create this mixture?

Show answer & explanation

Answer: 20

Answer

The distributor should use 20 kilograms of Blend A.

By setting up a system of linear equations based on the percentage of each bean type in Blend A (

A

) and Blend B (

B

), we get

0.60A + 0.20B = 14

and

0.40A + 0.80B = 16

. Solving this system yields

A = 20

and

B = 10

. Therefore, 20 kilograms of Blend A are required.

Step-by-Step Solution

1

Define variables for the unknowns and write the system of equations.

Let

A

be the number of kilograms of Blend A, and let

B

be the number of kilograms of Blend B. The system of equations is:
For Colombian beans:

0.60A + 0.20B = 14

For Ethiopian beans:

0.40A + 0.80B = 16

This translates the verbal and tabular constraints of the problem into mathematical equations.

2

Simplify the system by multiplying both sides of each equation to eliminate decimals.

Multiply the first equation by 5:

3A + B = 70 \implies B = 70 - 3A

Multiply the second equation by 5:

2A + 4B = 80 \implies A + 2B = 40

Working with integers makes the algebraic manipulation easier and reduces arithmetic errors.

3

Substitute the expression for

B

into the simplified second equation to solve for

A

.

A + 2(70 - 3A) = 40

A + 140 - 6A = 40

-5A = -100

A = 20

This isolates the variable

A

, which represents the required kilograms of Blend A.

Key Concept

Systems of Linear Equations

Question 172Question →

For a constant $k$ , the inequality $\frac{k - 3x}{2} > \frac{5x + 3}{4}$ has exactly 4 positive integer solutions for $x$ . If $k$ is an integer, how many possible values of $k$ are there?

Show answer & explanation

Answer: 6

Answer

The number of possible integer values for

k

is 6.

Simplifying the inequality gives

x < \frac{2k - 3}{11}

. For this inequality to have exactly 4 positive integer solutions, the solution set must contain only the integers 1, 2, 3, and 4. This requires the boundary

\frac{2k - 3}{11}

to satisfy

4 < \frac{2k - 3}{11} \le 5

. Solving for

k

gives

23.5 < k \le 29

. The integers in this interval are 24, 25, 26, 27, 28, and 29, which total 6 values.

Step-by-Step Solution

1

Multiply both sides of the inequality by 4.

2(k - 3x) > 5x + 3

Clearing the denominators makes the inequality easier to solve.

2

Distribute the 2 on the left side.

2k - 6x > 5x + 3

Expanding terms allows us to group variables.

3

Add

6x

to both sides and subtract 3 from both sides.

11x < 2k - 3

Grouping

x

on one side and parameter terms on the other side.

4

Divide by 11.

x < \frac{2k - 3}{11}

Isolating

x

gives the boundary for the solution set.

5

Establish the range for the boundary term

\frac{2k - 3}{11}

.

4 < \frac{2k - 3}{11} \le 5

For the inequality to contain exactly the positive integers 1, 2, 3, and 4, the boundary must lie strictly above 4 and less than or equal to 5.

6

Solve the compound inequality for

k

.

23.5 < k \le 29

Performing algebraic operations on all parts of the inequality to isolate

k

.

7

Identify and count all integer solutions for

k

.

6 integers (24, 25, 26, 27, 28, 29)

Counting the integers in the range

(23.5, 29]

yields the final answer.

Key Concept

Solving linear inequalities in one variable with parameter constraints and identifying integer solution sets.

Estimated Time:3m 0s

Question 173Question →

For all $x > 4$ , the expression $\frac{x^2 - 4x}{x - 2\sqrt{x}} + \frac{x\sqrt{x} + 8}{\sqrt{x} + 2}$ can be written in the form $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: 6

Answer

6

Factoring the numerator of the first term yields

x(\sqrt{x}-2)(\sqrt{x}+2)

and its denominator yields

\sqrt{x}(\sqrt{x}-2)

. Simplifying this term gives

x + 2\sqrt{x}

. Factoring the numerator of the second term as a sum of cubes gives

(\sqrt{x}+2)(x - 2\sqrt{x} + 4)

, which simplifies to

x - 2\sqrt{x} + 4

. Summing both simplified terms results in

2x + 4

. Matching this to the form

ax+b

gives

a=2

and

b=4

, so

a+b=6

.

Step-by-Step Solution

1

Simplify the first term of the expression.

\frac{x^2 - 4x}{x - 2\sqrt{x}} = x + 2\sqrt{x}

Factor

x

from the numerator to get

x(x-4)

and

\sqrt{x}

from the denominator to get

\sqrt{x}(\sqrt{x}-2)

. Rewrite

x-4

as the difference of squares

(\sqrt{x}-2)(\sqrt{x}+2)

, then cancel the common factor

\sqrt{x}-2

and simplify

\frac{x}{\sqrt{x}}

to

\sqrt{x}

.

2

Simplify the second term of the expression.

\frac{x\sqrt{x} + 8}{\sqrt{x} + 2} = x - 2\sqrt{x} + 4

Recognize

x\sqrt{x} + 8

as a sum of cubes,

(\sqrt{x})^3 + 2^3

. Factor it as

(\sqrt{x}+2)(x - 2\sqrt{x} + 4)

and cancel the common factor of

\sqrt{x}+2

in the denominator.

3

Add the simplified terms together.

2x + 4

Combine

(x + 2\sqrt{x})

and

(x - 2\sqrt{x} + 4)

by grouping like terms: the

2\sqrt{x}

and

-2\sqrt{x}

cancel out, leaving

2x + 4

.

4

Identify the values of

a

and

b

and find

a+b

.

6

Comparing

2x + 4

to

ax + b

gives

a = 2

and

b = 4

. Therefore,

a + b = 2 + 4 = 6

.

Key Concept

Simplifying rational expressions involving radicals by factoring (difference of squares and sum of cubes).

Question 174Question →

An oceanographic research vessel measures hydrostatic pressure during deep-sea dives. During a dive in temperate waters, the pressure $P$ , in atmospheres (atm), at a depth of $d$ meters below the surface is modeled by the linear equation $P = \rho d + 1.0$ , where $\rho$ is a constant representing the rate of change of pressure with respect to depth. At a depth of $150$ meters, the pressure is $16.0$ atm. During a second dive in a high-salinity region, the rate of change of pressure with respect to depth is $25\%$ greater than the rate in the temperate waters. According to the model for the second dive, at what depth, in meters, is the pressure $26.0$ atm?

Show answer & explanation

Answer: 200

Answer

200

To find the depth on the second dive where the pressure is

26.0

atm, we first find the slope of the linear model for the first dive. Given

P = \rho d + 1.0

, substituting the point

(150, 16.0)

gives

16.0 = 150\rho + 1.0

, which yields

\rho = 0.1

atm/m. The slope for the second dive is

25\%

greater, so the new slope is

0.1 \times 1.25 = 0.125

atm/m. The equation for the second dive is

P = 0.125d + 1.0

. Substituting

P = 26.0

yields

26.0 = 0.125d + 1.0

. Solving for

d

gives

25.0 = 0.125d

, which results in a depth of

200

meters.

Step-by-Step Solution

1

Find the rate of change of pressure with respect to depth in temperate waters.

\rho = 0.1

atm/m

Substitute the depth

d = 150

and pressure

P = 16.0

into the linear equation

P = \rho d + 1.0

to get

16.0 = 150\rho + 1.0

. Solving for

\rho

yields

15.0 = 150\rho

, which means

\rho = 0.1

.

2

Calculate the rate of change of pressure with respect to depth for the second dive.

\rho_{\text{new}} = 0.125

atm/m

The rate of change for the second dive is

25\%

greater than the rate of

0.1

atm/m from the first dive. Thus,

\rho_{\text{new}} = 0.1 \times 1.25 = 0.125

.

3

Calculate the depth where the pressure is

26.0

atm using the new rate of change.

d = 200

meters

Set up the equation for the second dive:

P = 0.125d + 1.0

. Substitute

P = 26.0

to get

26.0 = 0.125d + 1.0

. Subtracting

1.0

from both sides gives

25.0 = 0.125d

. Solving for

d

gives

d = \frac{25.0}{0.125} = 200

.

Key Concept

Interpreting and manipulating slope in a linear relationship context

Question 175Question →

A scientist is monitoring the volume of liquid nitrogen in a storage tank. The volume $V$ , in liters, of liquid nitrogen remaining in the tank $t$ hours after a cooling system malfunction is modeled by the equation $V = 240 - 1.5t$ . After how many hours of malfunction will there be exactly $180$ liters of liquid nitrogen remaining in the tank?

Show answer & explanation

Answer: 40

Answer

The cooling system malfunction must last 40 hours for there to be exactly 180 liters of liquid nitrogen remaining in the tank.

To find the number of hours of malfunction until

180

liters of liquid nitrogen remain in the tank, substitute

180

for

V

in the given model equation, yielding

180 = 240 - 1.5t

. Subtracting

240

from both sides of the equation results in

-60 = -1.5t

. Dividing both sides by

-1.5

gives the final value of

t = 40

hours.

Step-by-Step Solution

1

Substitute the target volume of liquid nitrogen into the linear equation.

180 = 240 - 1.5t

The question asks for the time

t

when the remaining volume

V

is exactly

180

liters.

2

Isolate the term containing the variable by subtracting the initial constant volume from both sides.

-60 = -1.5t

Subtracting

240

from both sides of the equation begins the process of isolating the variable

t

.

3

Divide both sides of the equation by the rate coefficient to solve for time.

t = 40

Dividing by

-1.5

isolates the variable

t

and gives the solution in hours.

Key Concept

Interpreting values and solving equations in linear contexts

Estimated Time:45s

Question 176Question →

For all $x \neq -2.5$ , the expression $\frac{6x^2 + 17x + 8}{2x + 5}$ is equivalent to $3x + 1 + \frac{a}{2x + 5}$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 3

Answer

The value of

a

is

3

.

Multiplying both sides of the equivalent relation by

2x + 5

yields

6x^2 + 17x + 8 = (3x + 1)(2x + 5) + a

. Expanding the right side gives

6x^2 + 17x + 5 + a

. Since the expressions are equivalent, the constant terms must be equal:

8 = 5 + a

, which simplifies to

a = 3

.

Step-by-Step Solution

1

Multiply both sides of the equation by

2x + 5

to clear the denominators.

6x^2 + 17x + 8 = (3x + 1)(2x + 5) + a

This clears the rational expressions so we can work with polynomials directly.

2

Expand the expression

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

using the distributive property.

6x^2 + 17x + 8 = 6x^2 + 15x + 2x + 5 + a

To write the right side as a polynomial in standard form.

3

Combine like terms on the right side of the equation.

6x^2 + 17x + 8 = 6x^2 + 17x + (5 + a)

To group coefficients of like powers of

x

for easy comparison.

4

Equate the constant terms from both sides of the equation to solve for

a

.

8 = 5 + a

, which gives

a = 3

.

For the two polynomial expressions to be equivalent for all values of

x

, their corresponding coefficients and constants must be equal.

Key Concept

Rewriting rational expressions by clearing denominators or polynomial long division to find equivalent expressions.

Question 177Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has the property that the value of $y$ decreases by $3$ for every increase of $5$ in the value of $x$ . If the graph of this equation passes through the point $(4, -2)$ and $a = 6$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is

4

.

The correct answer is

4

. The slope of the line described is

m = \frac{\Delta y}{\Delta x} = \frac{-3}{5} = -0.6

. Rearranging

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows that the slope is

-\frac{a}{b}

. Setting these equal gives

-\frac{a}{b} = -\frac{3}{5} \implies \frac{a}{b} = \frac{3}{5}

. Since

a = 6

, we find

b = 10

. Substituting

a = 6

,

b = 10

, and the coordinates of the point

(4, -2)

into the original equation yields

6(4) + 10(-2) = 4

.

Step-by-Step Solution

1

Determine the slope of the line from the described relationship between

x

and

y

.

The slope

m

of the line is

-\frac{3}{5}

.

The slope represents the change in

y

divided by the change in

x

. Since

y

decreases by

3

(

\Delta y = -3

) for every increase of

5

in

x

(

\Delta x = 5

), the slope is

m = \frac{\Delta y}{\Delta x} = -\frac{3}{5}

.

2

Express the slope of the line in terms of the coefficients from the standard form equation

ax + by = c

.

The slope of the line is

-\frac{a}{b}

.

Rewriting the equation

ax + by = c

in slope-intercept form gives

by = -ax + c \implies y = -\frac{a}{b}x + \frac{c}{b}

. The coefficient of

x

is the slope, so

m = -\frac{a}{b}

.

3

Equate the two expressions for the slope and solve for

b

using the given value of

a = 6

.

b = 10

Setting

-\frac{a}{b} = -\frac{3}{5}

gives

\frac{a}{b} = \frac{3}{5}

. Substituting

a = 6

yields

\frac{6}{b} = \frac{3}{5}

, which simplifies to

3b = 30

, so

b = 10

.

4

Substitute the point

(4, -2)

and the values of

a

and

b

into the equation

ax + by = c

to solve for

c

.

c = 4

Substituting

a = 6

,

b = 10

,

x = 4

, and

y = -2

into

ax + by = c

gives

6(4) + 10(-2) = c \implies 24 - 20 = c \implies c = 4

.

Key Concept

Linear Equations in Two Variables

Question 178Question →

A bakery sells blueberry muffins and bran muffins. The total cost of $6$ blueberry muffins and $4$ bran muffins is $\$22.00$ . The total cost of $5$ blueberry muffins and $8$ bran muffins is $\$26.50$ . What is the cost, in dollars, of $1$ blueberry muffin?

Show answer & explanation

Answer: 2.5

Answer

The cost of one blueberry muffin is 2.5 dollars.

The correct answer is 2.5. By setting up the system of linear equations

6x + 4y = 22

and

5x + 8y = 26.5

, where

x

represents the cost of one blueberry muffin and

y

represents the cost of one bran muffin, we can solve for

x

. Multiplying the first equation by 2 gives

12x + 8y = 44

. Subtracting the second equation,

5x + 8y = 26.5

, from the modified first equation yields

7x = 17.5

. Dividing by 7 results in

x = 2.5

.

Step-by-Step Solution

1

Set up a system of equations.

6x + 4y = 22

and

5x + 8y = 26.5

Translate the context into mathematical statements where

x

is the price of a blueberry muffin and

y

is the price of a bran muffin.

2

Multiply the first equation by 2.

12x + 8y = 44

Align the coefficients of

y

to facilitate elimination.

3

Subtract the second equation from the modified first equation.

7x = 17.5

Eliminate the variable

y

to solve for

x

.

4

Divide by 7.

x = 2.5

Isolate the variable

x

to find the cost of a blueberry muffin.

Key Concept

Solving systems of two linear equations using elimination.

Alternative Method

Alternatively, solve for

y

in terms of

x

using the first equation:

y = 5.5 - 1.5x

. Substitute this expression into the second equation:

5x + 8(5.5 - 1.5x) = 26.5

, which simplifies to

5x + 44 - 12x = 26.5 \\Rightarrow -7x = -17.5 \\Rightarrow x = 2.5

.

Estimated Time:1m 30s

Question 179Question →

A linear equation and a quadratic equation form a system, as shown.

y = x^2 + 5

y = 3x + 5

If the ordered pair $(x, y)$ is a solution to the system where $x$ is positive, what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Equating the two expressions for

y

gives the equation

x^2 + 5 = 3x + 5

. Subtracting 5 from both sides results in

x^2 = 3x

. Subtracting

3x

from both sides gives the quadratic equation

x^2 - 3x = 0

, which can be factored as

x(x - 3) = 0

. This yields two solutions for

x

:

0

and

3

. Since the question specifies that

x

is positive, the correct value is 3.

Step-by-Step Solution

1

Equate the equations

x^2 + 5 = 3x + 5

Since both equations are solved for

y

, we can substitute the quadratic expression into the linear equation.

2

Simplify the equation

x^2 - 3x = 0

Subtracting 5 from both sides and then subtracting

3x

from both sides collects all terms on one side.

3

Factor and solve

x(x - 3) = 0

, so

x = 0

or

x = 3

Factoring out the greatest common factor,

x

, allows us to apply the zero product property to find the individual roots.

4

Apply the constraint

x = 3

The problem states that

x

must be positive, which excludes the solution

x = 0

.

Key Concept

Solving a system consisting of a linear equation and a quadratic equation by substitution.

Question 180Question →

If the expression $\frac{10x^2 + x - 3}{2x - 1}$ is equivalent to $ax + b$ for all $x \neq 0.5$ , where $a$ and $b$ are constants, what is the value of $a + b$ ?

Show answer & explanation

Answer: 8

Answer

The value of

a + b

is 8.

To find the equivalent expression, factor the numerator:

10x^2 + x - 3 = (2x - 1)(5x + 3)

. For all

x \neq 0.5

, the denominator

2x - 1

is non-zero, so we can divide out the common factor

(2x - 1)

to get

5x + 3

. Matching

5x + 3

with

ax + b

gives

a = 5

and

b = 3

. The sum of these constants is

5 + 3 = 8

.

Step-by-Step Solution

1

Factor the numerator of the expression

10x^2 + x - 3 = (2x - 1)(5x + 3)

Factoring the quadratic trinomial allows us to identify common factors that can be simplified.

2

Simplify the rational expression

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

(for

x \neq 0.5

)

Since

x \neq 0.5

, the term

2x - 1

is non-zero and can be canceled from both the numerator and the denominator.

3

Identify the values of

a

and

b

a = 5

and

b = 3

By comparing the simplified expression

5x + 3

to the form

ax + b

, the coefficients of corresponding terms must be equal.

4

Calculate the sum of

a

and

b

5 + 3 = 8

The question asks for the sum of the constants

a

and

b

.

Key Concept

Simplifying rational expressions by factoring and coefficient matching

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer