Question

Put the steps in order that are used to solve the following systems of equations by substitution.

−7x−2y=−13

x−2y=11
(10 points)

responses
-16y - 77 = -13
-7 (2y+11) - 2y = -13
-16y = 64
x - 2y = 11 --> x = 2y + 11
x- 2 (-4)=11
-14y - 77 - 2y = -13
(3, -4)
x + 8 = 11
x = 3
y = -4

Answer 1

A company owns two dealerships, both of which sell cars and trucks. Dealership A sells a total of 164 cars and trucks. Dealership B sells twice as many cars and half as many trucks as Dealership A, and sells a total of 229 cars and trucks.

(4 points)
a) An equation for the total cars and trucks for Dealership A: (response area 1)

b) An equation for the total cars and trucks for Dealership B: (response area 2)

c) How many cars did Dealership A sell? (response area 3)

d) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell? (response area 4)

1 responses
x + y = 229
x + y = 393
x + y = 164

2 responses
1/2x + 1/2y = 229
2x + 1/2y = 229
1/2x + 2y = 164

3 responses
132
98
196
66

4 responses
33
66
49
98

Answer 2

To solve the system of equations by substitution, the steps can be arranged in the following order:

1. **Start with the second equation and solve for x:**
- \( x - 2y = 11 \) → \( x = 2y + 11 \)

2. **Substitute x into the first equation:**
- \(-7(2y + 11) - 2y = -13\)

3. **Distribute and simplify the equation:**
- \(-14y - 77 - 2y = -13\) → \(-16y - 77 = -13\)

4. **Combine like terms:**
- \(-16y = 64\)

5. **Solve for y:**
- \(y = -4\)

6. **Substitute y back into the equation for x:**
- \(x - 2(-4) = 11\)

7. **Solve for x:**
- \(x + 8 = 11\) → \(x = 3\)

8. **State the final solution as an ordered pair:**
- \((3, -4)\)

Putting this all together, the ordered steps are:

1. \(x - 2y = 11 \Rightarrow x = 2y + 11\)
2. Substitute: \(-7(2y + 11) - 2y = -13\)
3. Distribute: \(-14y - 77 - 2y = -13\)
4. Combine terms: \(-16y - 77 = -13\)
5. Solve for y: \(-16y = 64\)
6. Substitute: \(x - 2(-4) = 11\)
7. Solve for x: \(x + 8 = 11\)
8. Final solution: \((3, -4)\)

Answer 3

To solve this problem, we'll first define \( x \) as the number of cars sold at Dealership A and \( y \) as the number of trucks sold at Dealership A.

### a) An equation for the total cars and trucks for Dealership A:
Since Dealership A sells a total of 164 cars and trucks, the equation is:
- **Response Area 1:** \( x + y = 164 \)

### b) An equation for the total cars and trucks for Dealership B:
Dealership B sells twice as many cars as Dealership A could be expressed as \( 2x \) and half as many trucks as Dealership A could be expressed as \( \frac{1}{2}y \). Since the total sales at Dealership B is 229 cars and trucks, the equation is:
- **Response Area 2:** \( 2x + \frac{1}{2}y = 229 \)

### c) How many cars did Dealership A sell?
We can solve the system of equations:
1. \( x + y = 164 \) (Equation for Dealership A)
2. \( 2x + \frac{1}{2}y = 229 \) (Equation for Dealership B)

From the first equation, we can express \( y \) in terms of \( x \):
- \( y = 164 - x \)

Now substitute \( y \) into the second equation:
- \( 2x + \frac{1}{2}(164 - x) = 229 \)
- Simplifying gives:
\( 2x + 82 - \frac{1}{2}x = 229 \)
\( \frac{4x - x}{2} + 82 = 229 \)
\( \frac{3x}{2} + 82 = 229 \)
\( \frac{3x}{2} = 229 - 82 \)
\( \frac{3x}{2} = 147 \)
\( 3x = 294 \)
\( x = 98 \)

So Dealership A sold:
- **Response Area 3:** \( 98 \)

### d) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
Since Dealership B sells half the number of trucks as Dealership A, we will first find the number of trucks sold by Dealership A:
- From \( y = 164 - x \)
- \( y = 164 - 98 = 66 \) (trucks sold at Dealership A)

Dealership B sells half of this:
- Trucks sold by Dealership B: \( \frac{1}{2}y = \frac{1}{2}(66) = 33 \)

So, Dealership B sold:
- **Response Area 4:** \( 33 \)

### Summary of Responses:
- **Response Area 1:** \( x + y = 164 \)
- **Response Area 2:** \( 2x + \frac{1}{2}y = 229 \)
- **Response Area 3:** \( 98 \) (cars sold by Dealership A)
- **Response Area 4:** \( 33 \) (trucks sold by Dealership B)