Question 1

Graph the solution set of the following system:

Accepted Answer

The answer of Graph the solution set of the following...

Question 2

Solve by the addition method:

Accepted Answer

It is impossible to solve the equation with the given information. The provided expression does not represent a system of equations or any equation at all.

Question 3

A chemist has two alloys, one of which is 10% gold and 20% lead, and the other of which is 30% gold and 40% lead. How many grams of each of the two alloys should be used to make an alloy that contains 65 g of gold and 118 g of lead?

Accepted Answer

Let x be the amount of the first alloy and y be the amount of the second alloy.

We can set up a system of two equations to represent the problem:

0.1x + 0.3y = 65   (the equation for the amount of gold)
0.2x + 0.4y = 118  (the equation for the amount of lead)

To solve for x and y, we can use any method of solving systems of equations. One common method is to use elimination.

We can multiply the first equation by 2 and subtract it from the second equation to eliminate y:

0.2x + 0.4y = 118
-0.2x - 0.6y = -130
-------------------
0.0x - 0.2y = -12

Solving for y, we get:

y = 60

Substituting y back into either of the original equations, we can solve for x:

0.1x + 0.3y = 65
0.1x + 0.3(60) = 65
0.1x + 18 = 65
0.1x = 47
x = 470

Therefore, we need 470 grams of the first alloy and 60 grams of the second alloy to make an alloy with 65 grams of gold and 118 grams of lead, so the answer is D.

Question 4

Rowing with the current, a canoeist paddled 14 mi in 2 h. Against the current, the canoeist could paddle only 6 mi in the same amount of time. Find the rate of the canoest in calm water and the rate of the current.

Accepted Answer

The rate of the canoeist in calm water is the average of the speeds going with and against the current: (14/2 + 6/2) / 2 = 5 mph. The rate of the current is the difference between the canoeist's rate in calm water and the rate against the current: 5 - (6/2) = 2 mph.

Question 5

Solve by the addition method:

Accepted Answer

The answer of Solve by the addition method: 
...

Question 6

An investor has a total of $24,000 deposited in three different accounts, which earn annual interest rates of 9%, 7%, and 5%. The amount deposited in the 9% account is twice the amount in the 7% account. If the three accounts earn a total annual interest of $1700, how much money is deposited in each account?

Accepted Answer

Let x be the amount in the 7% account. Then, 2x is in the 9% account, and the rest, $24,000 - 3x, is in the 5% account. The equation for interest is: 0.09(2x) + 0.07(x) + 0.05(24000 - 3x) = 1700. Solving gives x = $5000, 2x = $10,000, and $9000 in the 5% account.

Question 7

Graph the solution set of the following system:

Accepted Answer

The answer of Graph the solution set of the following...

Question 8

Solve by the addition method:

Accepted Answer

The answer of Solve by the addition method: 
...

Question 9

A relief organization supplies blankets, cots, and lanterns to victims of fires, floods, and other natural disasters. One week the organization purchased 10 blankets, 10 cots, and 20 lanterns for a total cost of $1800. The second week, at the same prices, the organization purchased 20 blankets, 20 cots, and 10 lanterns for a total cost of $2400. The third week, at the same prices, the organization purchased 20 blankets, 10 cots, and 10 lanterns for a total cost of $1650. Find the cost of one blanket, the cost of one cot, and the cost of one lantern.

Accepted Answer

Let x be the cost of one blanket, y be the cost of one cot, and z be the cost of one lantern. Then we can set up the following system of equations:
10x + 10y + 20z = 1800
20x + 20y + 10z = 2400
20x + 10y + 10z = 1650
Simplifying each equation by dividing by 10, we get:
x + y + 2z = 180
2x + 2y + z = 240
2x + y + z = 165
Solving this system of equations (e.g. by elimination or substitution), we get x = 25, y = 75, and z = 40. Therefore, the cost of one blanket is $25, the cost of one cot is $75, and the cost of one lantern is $40. Therefore, the answer is choice C.

Question 10

A baker purchased 14 lb of wheat flour and 16 lb of rye flour for a total cost of $21.72. A second purchase, at the same prices, included 14 lb of wheat flour and 13 lb of rye flour. The cost of the second purchase was $19.59. Find the cost per pound of the wheat flour and the rye flour.

Accepted Answer

Let the cost per pound of wheat be $w and the cost per pound of rye be $r. Then we have the following system of equations:

14w + 16r = 21.72
14w + 13r = 19.59

Subtracting the second equation from the first, we get:

3r = 2.13

Therefore, r = 0.71. Substituting this value into either equation, we can solve for w:

14w + 16(0.71) = 21.72
14w = 10.76
w = 0.77

Therefore, the cost per pound of wheat is $0.74 and the cost per pound of rye is $0.71. Choice C is the correct answer.

Question 11

Graph the solution set of the following system:

Accepted Answer

As there are two variables, the solution set will be a line in the Cartesian plane. By rearranging the equations, we get the equation: $y = \frac{1}{2}x + 3$. This is the equation of a line with slope $\frac{1}{2}$ and $y$-intercept $(0,3)$. Therefore, the solution set is a line that passes through $(0,3)$ and has a slope of $\frac{1}{2}$. The only option that fits this description is D.

Question 12

A plane flying with the jet stream flew from Los Angeles to Chicago, a distance of 2568 mi, in 4 h. Flying against the jet stream, the plane could fly only 2212 mi in the same amount of time. Find the rate of the plane in calm air and the rate of the wind.

Accepted Answer

The answer of A plane flying with the jet stream...

Question 13

Solve by the addition method:

Accepted Answer

The answer of Solve by the addition method: 
...

Question 14

Solve by the addition method:

Accepted Answer

Since the question only asks for the solution method and not the actual answer, we can use process of elimination to determine the correct choice.

To use the addition method, we need to add the two equations together in a way that eliminates one of the variables. Looking at the answer choices, we can see that choice E has a combination of coefficients that will allow us to add the equations together and eliminate the y variable.

11x + 8y + 3z = 109
x + 4y - z = -15

Multiplying the second equation by -2, we get:

-2x - 8y + 2z = 30

Now, when we add this to the first equation, the y variable cancels out:

9x + 5z = 139

This is a linear equation in two variables, so we can solve for either x or z using algebraic methods. Therefore, choice E is the best choice for this problem.

Question 15

Solve by the addition method:

Accepted Answer

The given equation contains hexadecimal digits (a-f), which are not valid in a decimal system. Therefore, the equation cannot be solved using the addition method, and there is no solution.

Question 16

Graph the solution set of the following system:

Accepted Answer

The answer of Graph the solution set of the following...

Quiz 5: Systems of Linear Equations and Inequalities

Graph the solution set of the following system:

Solve by the addition method:

A chemist has two alloys, one of which is 10% gold and 20% lead, and the other of which is 30% gold and 40% lead. How many grams of each of the two alloys should be used to make an alloy that contains 65 g of gold and 118 g of lead?

Rowing with the current, a canoeist paddled 14 mi in 2 h. Against the current, the canoeist could paddle only 6 mi in the same amount of time. Find the rate of the canoest in calm water and the rate of the current.

Solve by the addition method:

Graph the solution set of the following system:

Solve by the addition method:

A baker purchased 14 lb of wheat flour and 16 lb of rye flour for a total cost of $21.72. A second purchase, at the same prices, included 14 lb of wheat flour and 13 lb of rye flour. The cost of the second purchase was $19.59. Find the cost per pound of the wheat flour and the rye flour.

Graph the solution set of the following system:

A plane flying with the jet stream flew from Los Angeles to Chicago, a distance of 2568 mi, in 4 h. Flying against the jet stream, the plane could fly only 2212 mi in the same amount of time. Find the rate of the plane in calm air and the rate of the wind.

Solve by the addition method:

Solve by the addition method:

Solve by the addition method:

Graph the solution set of the following system: