Question 1

Graph the solution of the system: $$\left\{ \begin{array} { l } 
y < - 4 x - 1 \
y > - 4 x + 2
\end{array} \right.$$

Accepted Answer

The answer of Graph the solution of the system: \[\left\{...

Question 2

An airplane can fly with the wind a distance of 920 miles in 4 hours. However, the return trip against the wind takes 5 hours. Find the speed of the plane in still air and the speed of the wind.

Accepted Answer

Let's call the speed of the plane in still air "p" and the speed of the wind "w". Using the formula distance = rate x time, we can create two equations based on the information given:

With the wind: 920 = (p + w) x 4
Against the wind: 920 = (p - w) x 5

Simplifying both equations, we get:

p + w = 230
p - w = 184

Adding the two equations together eliminates the "w" variable, giving us:

2p = 414
p = 207

Substituting p back into one of the equations, we can find the value of w:

p + w = 230
207 + w = 230
w = 23

Therefore, the speed of the plane in still air is 207 mph and the speed of the wind is 23 mph. The answer is D.

Question 3

Use two equations in two variables to solve the following problem. Two pairs of shoes and four pairs of socks cost $107, and three pairs of shoes and five pairs of socks cost $156. Find the cost of a pair of socks.

Accepted Answer

The answer of Use two equations in two variables to...

Question 4

A merchant wishes to mix peanuts that sell for $2.5 per pound and cashews that sell for $8 per pound to get 100 pounds of mixed nuts that sell for $4.50 per pound. Let x equal the number of pounds of peanuts and y equal the number of pounds of cashews. If solving this by writing two equations in two variables, which of the following could be one of the equations?

Accepted Answer

The equation should represent the total value of the mixture, which is the sum of the values of the peanuts and cashews. We can represent this as:

$2.5x + $8y = total value of nuts

We want the total value of nuts to be $4.50 per pound, so we also know that:

total value of nuts / 100 = $4.50

Simplifying this, we get:

total value of nuts = $450

Substituting into the first equation, we get:

$2.5x + $8y = $450

So the correct equation is A, which represents the total value of the mixture. None of the other choices represent the total value, and B is incorrect because it sets the total value equal to the price per pound instead of solving for it.

Question 5

Graph the solution of the system: $$\left\{ \begin{array} { l } 
y < 5 x - 3 \
y - 5 x \geq 4
\end{array} \right.$$

Accepted Answer

The first inequality $y<5x-3$ represents a line with a slope of 5 passing through the point $(0,-3)$ and shaded below the line.

The second inequality $y-5x\geq4$ represents a line with a slope of 5 passing through the point $(0,4)$ and shaded above the line.

The graph of the system is the region that satisfies both inequalities, therefore, it is the shaded region below the green line ($y<5x-3$) and above the red line ($y-5x\geq4$):

![Graph of the system](https://i.ibb.co/1nJS5rc/system.png)

Question 6

A boat traveled 48 miles downstream in 4 hours and made the return trip in 6 hours. Find the rate of the current.

Accepted Answer

The rate of the current is 2 mph. This is found by first determining the speed of the boat in still water and the effect of the current. Downstream speed is (48 miles / 4 hours) = 12 mph, and upstream speed is (48 miles / 6 hours) = 8 mph. The current's speed is the difference between the boat's speed in still water and its speed with the current, divided by 2. Thus, (12 mph - 8 mph) / 2 = 2 mph.

Question 7

Part of $2,200 is invested at 12%, another part at 13%, and the remainder at 14%. The total yearly income from the three investments is $295. The sum of the amounts invested at 12% and 13% equals the amount invested at 14%. Determine how much is invested at each rate.

Accepted Answer

The answer of Part of $2,200 is invested at 12%,...

Question 8

Melodic Music has compact discs on sale for either $15 or $20. If a customer wants to spend at least $60 but no more than $120 on CDs, graph a system of inequalities that will show the possible ways a customer can buy $15 CDs ( x )and $20 CDs ( y ).

Accepted Answer

The answer of Melodic Music has compact discs on sale...

Question 9

Dry Boat Works wholesales aluminum boats for $500 and fiberglass boats for $300. Northland Marina wants to order at least $750 worth but no more than $1,500 worth of boats. Write a system of linear inequalities which describes all possible combinations of aluminum boats ( x )and fiberglass boats ( y )that can be ordered, graph the system of inequalities, and determine whether the ordered pairs satisfy the inequality.

Accepted Answer

We want to find the system of inequalities that defines the feasible region for Northland Marina's boat order. Let x be the number of aluminum boats and y be the number of fiberglass boats.

From the problem, we know that the cost of ordering x aluminum boats and y fiberglass boats should be at least $750 but no more than $1500.

Therefore, we can construct the following system of linear inequalities: 
\begin{align*}
500x + 300y &\leq 1500 \
500x + 300y &\geq 750
\end{align*}

These inequalities represent the feasible region for Northland Marina's order. To graph this system of inequalities, we can rewrite them in slope-intercept form: 
\begin{align*}
y &\leq -\frac{5}{3} x + 5 \
y &\geq -\frac{5}{3} x + \frac{5}{2}
\end{align*}

The graph of these inequalities is a shaded region bounded by the lines y = -5/3x + 5 and y = -5/3x + 5/2. See the graph below:

![Graph of System of Inequalities](https://i.imgur.com/kUpiH74.png)

To determine whether the ordered pairs (1,2) and (3,0) satisfy the inequality, we can plug them into the inequalities: 
\begin{align*}
500(1) + 300(2) &= 1100 \quad 	ext{which satisfies both inequalities} \
500(3) + 300(0) &= 1500 \quad 	ext{which satisfies the first inequality, but not the second}
\end{align*}

Therefore, the feasible region is the shaded region in the graph, and the best choice for the system of inequalities is option C.

Question 10

In a mix of pennies and nickels, there are 22 coins with a total value of 82 cents. How many nickels are there?

Accepted Answer

The answer of In a mix of pennies and nickels,...

Question 11

Find the solution set of the system of inequalities. $$\left\{ \begin{array} { l } 
x + y < - 6 \
x - y > - 6
\end{array} \right.$$

Accepted Answer

The answer of Find the solution set of the system...

Question 12

Find the solution set of the system of inequalities. $$\left\{ \begin{array} { c } 
x - y \geq 6 \
x + 2 y < - 4
\end{array} \right.$$

Accepted Answer

The answer of Find the solution set of the system...

Question 13

Doris invested some money at 7% and some money at 8%. She invested $4,000 more at 8% than she did at 7%. Her total yearly interest from the two investments was $920. How much did Doris invest at each rate?

Accepted Answer

Let x be the amount invested at 7%. Then the amount invested at 8% is x + $4,000.

The interest from the 7% investment is 0.07x, and the interest from the 8% investment is 0.08(x + $4,000), since that investment is $4,000 more.

The total interest is $920. So we can write an equation:

0.07x + 0.08(x + $4,000) = $920

Simplifying:

0.07x + 0.08x + $320 = $920

0.15x = $600

x = $4,000

So Doris invested $4,000 at 7% and $8,000 ($4,000 + $4,000) at 8%. The answer is D.

Question 14

Find the solution set of the system of inequalities. $$\left\{ \begin{array} { c } 
5 x + 3 y > - 8 \
4 x - 5 y \geq 1
\end{array} \right.$$

Accepted Answer

The answer of Find the solution set of the system...

Question 15

Graph the solution of the system: $$\left\{ \begin{array} { c } 
x - 2 y < 4 \
y < \frac { x } { 2 } - 5
\end{array} \right.$$

Accepted Answer

The answer of Graph the solution of the system: \[\left\{...

Question 16

Use two equations in two variables to solve the following problem. The salaries of the president and vice president of the United States total $490,000 a year. If the President makes $210,000 more than the Vice president, find each of their salaries.

Accepted Answer

Let P be the President's salary and V be the Vice President's salary. We have two equations: P + V = 490,000 and P = V + 210,000. Solving these, we get V = 140,000 and P = 350,000.

Question 17

A distributor wholesales desk chairs for $200 and side chairs for $100. Best Furniture wants to order no more than $1,200 worth of chairs and wants to order more side chairs than desk chairs. Find a system of inequalities whose graph will show the possible combination of desk chairs ( x )and side chairs ( y )that can be ordered, graph the system of inequalities, and determine whether the ordered pairs satisfy the inequality.

Accepted Answer

The answer of A distributor wholesales desk chairs for $200...

Question 18

One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 45 miles in 5 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 27 miles in 1 hour. Find the rate of the current.

Accepted Answer

The rate of the current can be found by setting up equations based on the information given. Let $x$ be Jim's kayaking speed in still water and $y$ be the speed of the current. When paddling upstream, his effective speed is $x - y$, and when paddling downstream, it's $x + y$. From the information given: $45 = 5(x - y)$ and $27 = 1(2x + 2y)$. Solving these equations simultaneously gives $x = 12$ mph and $y = 3$ mph.

Question 19

Part of $10,000 is invested at 9%, another part at 10%, and the remainder at 12% yearly interest. The total yearly income from the three investments is $1,075. The sum of the amounts invested at 9% and 10% equals the amount invested at 12%. How much is invested at each rate?

Accepted Answer

The total investment is $10,000, and the sum invested at 9% and 10% equals the amount invested at 12%, which means each of these portions is $5,000. The total interest is $1,075, which matches the distribution of $2,500 at 9%, $2,500 at 10%, and $5,000 at 12% when calculated.

Question 20

A gift store is making a mixture of almonds, pecans, and peanuts, which sell for $3.00 per pound, $4.00 per pound, and $2.00 per pound, respectively. The storekeeper wants to make 20 pounds of the mix to sell at $2.70 per pound. The number of pounds of peanuts is to be three times the number of pounds of pecans. Find the number of pounds of each to be used in the mixture. # of almonds = __________ # of pecans = __________ # of peanuts = __________

Accepted Answer

The answer of A gift store is making a mixture...

Quiz 4: Solving Systems of Equations and Inequalities

Graph the solution of the system: $\left\{ \begin{array} { l } y < - 4 x - 1 \\y > - 4 x + 2\end{array} \right.$

An airplane can fly with the wind a distance of 920 miles in 4 hours. However, the return trip against the wind takes 5 hours. Find the speed of the plane in still air and the speed of the wind.

Use two equations in two variables to solve the following problem. Two pairs of shoes and four pairs of socks cost $107, and three pairs of shoes and five pairs of socks cost $156. Find the cost of a pair of socks.

Graph the solution of the system: $\left\{ \begin{array} { l } y < 5 x - 3 \\y - 5 x \geq 4\end{array} \right.$

A boat traveled 48 miles downstream in 4 hours and made the return trip in 6 hours. Find the rate of the current.

Part of $2,200 is invested at 12%, another part at 13%, and the remainder at 14%. The total yearly income from the three investments is $295. The sum of the amounts invested at 12% and 13% equals the amount invested at 14%. Determine how much is invested at each rate.

Melodic Music has compact discs on sale for either $15 or $20. If a customer wants to spend at least $60 but no more than $120 on CDs, graph a system of inequalities that will show the possible ways a customer can buy $15 CDs ( x ) and $20 CDs ( y ) .

In a mix of pennies and nickels, there are 22 coins with a total value of 82 cents. How many nickels are there?

Find the solution set of the system of inequalities. $\left\{ \begin{array} { l } x + y < - 6 \\x - y > - 6\end{array} \right.$

Find the solution set of the system of inequalities. $\left\{ \begin{array} { c } x - y \geq 6 \\x + 2 y < - 4\end{array} \right.$

Doris invested some money at 7% and some money at 8%. She invested $4,000 more at 8% than she did at 7%. Her total yearly interest from the two investments was $920. How much did Doris invest at each rate?

Find the solution set of the system of inequalities. $\left\{ \begin{array} { c } 5 x + 3 y > - 8 \\4 x - 5 y \geq 1\end{array} \right.$

Graph the solution of the system: $\left\{ \begin{array} { c } x - 2 y < 4 \\y < \frac { x } { 2 } - 5\end{array} \right.$

Use two equations in two variables to solve the following problem. The salaries of the president and vice president of the United States total $490,000 a year. If the President makes $210,000 more than the Vice president, find each of their salaries.

Part of $10,000 is invested at 9%, another part at 10%, and the remainder at 12% yearly interest. The total yearly income from the three investments is $1,075. The sum of the amounts invested at 9% and 10% equals the amount invested at 12%. How much is invested at each rate?

Quiz 4: Solving Systems of Equations and Inequalities

Graph the solution of the system: {y<−4x−1y>−4x+2\left\{ \begin{array} { l } y < - 4 x - 1 \\y > - 4 x + 2\end{array} \right.{y<−4x−1y>−4x+2​

An airplane can fly with the wind a distance of 920 miles in 4 hours. However, the return trip against the wind takes 5 hours. Find the speed of the plane in still air and the speed of the wind.

Use two equations in two variables to solve the following problem. Two pairs of shoes and four pairs of socks cost $107, and three pairs of shoes and five pairs of socks cost $156. Find the cost of a pair of socks.

Graph the solution of the system: {y<5x−3y−5x≥4\left\{ \begin{array} { l } y < 5 x - 3 \\y - 5 x \geq 4\end{array} \right.{y<5x−3y−5x≥4​

A boat traveled 48 miles downstream in 4 hours and made the return trip in 6 hours. Find the rate of the current.

Part of $2,200 is invested at 12%, another part at 13%, and the remainder at 14%. The total yearly income from the three investments is $295. The sum of the amounts invested at 12% and 13% equals the amount invested at 14%. Determine how much is invested at each rate.

Melodic Music has compact discs on sale for either $15 or $20. If a customer wants to spend at least $60 but no more than $120 on CDs, graph a system of inequalities that will show the possible ways a customer can buy $15 CDs ( x ) and $20 CDs ( y ) .

In a mix of pennies and nickels, there are 22 coins with a total value of 82 cents. How many nickels are there?

Find the solution set of the system of inequalities. {x+y<−6x−y>−6\left\{ \begin{array} { l } x + y < - 6 \\x - y > - 6\end{array} \right.{x+y<−6x−y>−6​

Find the solution set of the system of inequalities. {x−y≥6x+2y<−4\left\{ \begin{array} { c } x - y \geq 6 \\x + 2 y < - 4\end{array} \right.{x−y≥6x+2y<−4​

Doris invested some money at 7% and some money at 8%. She invested $4,000 more at 8% than she did at 7%. Her total yearly interest from the two investments was $920. How much did Doris invest at each rate?

Find the solution set of the system of inequalities. {5x+3y>−84x−5y≥1\left\{ \begin{array} { c } 5 x + 3 y > - 8 \\4 x - 5 y \geq 1\end{array} \right.{5x+3y>−84x−5y≥1​

Graph the solution of the system: {x−2y<4y<x2−5\left\{ \begin{array} { c } x - 2 y < 4 \\y < \frac { x } { 2 } - 5\end{array} \right.{x−2y<4y<2x​−5​

Use two equations in two variables to solve the following problem. The salaries of the president and vice president of the United States total $490,000 a year. If the President makes $210,000 more than the Vice president, find each of their salaries.

Part of $10,000 is invested at 9%, another part at 10%, and the remainder at 12% yearly interest. The total yearly income from the three investments is $1,075. The sum of the amounts invested at 9% and 10% equals the amount invested at 12%. How much is invested at each rate?

Graph the solution of the system: $\left\{ \begin{array} { l } y < - 4 x - 1 \\y > - 4 x + 2\end{array} \right.$

Graph the solution of the system: $\left\{ \begin{array} { l } y < 5 x - 3 \\y - 5 x \geq 4\end{array} \right.$

Find the solution set of the system of inequalities. $\left\{ \begin{array} { l } x + y < - 6 \\x - y > - 6\end{array} \right.$

Find the solution set of the system of inequalities. $\left\{ \begin{array} { c } x - y \geq 6 \\x + 2 y < - 4\end{array} \right.$

Find the solution set of the system of inequalities. $\left\{ \begin{array} { c } 5 x + 3 y > - 8 \\4 x - 5 y \geq 1\end{array} \right.$

Graph the solution of the system: $\left\{ \begin{array} { c } x - 2 y < 4 \\y < \frac { x } { 2 } - 5\end{array} \right.$