Question

Given the following system of linear equations, do Part A and Part B below.
x+y=7
2x+y=5




Part A: Solve the system of linear equations using substitution.
Part B: Use a graph to verify your answer to the system.
In your final answer, include all of your work for both Parts A and B as well as the graph for Part B.

Answer 1

Let's solve the given system of linear equations step by step:

The equations are:

1. \(x + y = 7\)
2. \(2x + y = 5\)

### Part A: Solve the system of linear equations using substitution

#### Step 1: Solve one equation for one variable

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

\[
y = 7 - x
\]

#### Step 2: Substitute into the second equation

Now, substitute this expression for \(y\) into the second equation:

\[
2x + y = 5
\]
\[
2x + (7 - x) = 5
\]

#### Step 3: Simplify and solve for \(x\)

Next, we'll simplify the equation:

\[
2x + 7 - x = 5
\]
\[
x + 7 = 5
\]
Subtract 7 from both sides:

\[
x = 5 - 7
\]
\[
x = -2
\]

#### Step 4: Substitute back to find \(y\)

Now, use the value of \(x\) to find \(y\):

\[
y = 7 - (-2)
\]
\[
y = 7 + 2
\]
\[
y = 9
\]

#### Final solution for Part A:

The solution to the system of equations is:

\[
(x, y) = (-2, 9)
\]

### Part B: Use a graph to verify your answer

Now let's graph both equations to confirm the solution.

1. **Graphing the first equation \(x + y = 7\)**:
- Rewrite it in slope-intercept form \(y = -x + 7\).
- The y-intercept is 7 (point (0, 7)).
- The slope is -1 (for every 1 unit x increases, y decreases by 1).
- Another point can be found by plugging in \(x = 7\) (this gives \(y = 0\)): point (7, 0).

2. **Graphing the second equation \(2x + y = 5\)**:
- Rewrite it in slope-intercept form \(y = -2x + 5\).
- The y-intercept is 5 (point (0, 5)).
- The slope is -2 (for every 1 unit x increases, y decreases by 2).
- Another point can be found by plugging in \(x = 0\) (this gives \(y = 5\)) and \(x = 2.5\) (this gives \(y = 0\)): point (2.5, 0).

With these points, we can draw the two lines on a graph.

#### Example of the graph:

```plaintext
10 | *
|
9 | *
|
8 |
|
7 | *
|
6 |
|
5 | *
|
4 |
|
3 |
|
2 |
|
1 |
|
0 | *
|________________________
-2 -1 0 1 2 3 4 5 6 7
```

### Conclusion for Part B:

The intersection point of the two lines is at \((-2, 9)\), which confirms our solution from Part A. Thus, the final answer is:

\[
\text{Part A: } (x, y) = (-2, 9)
\]
\[
\text{Part B: Verified using graph with intersection at (-2, 9).}
\]

You can plot these points and lines on a graphing tool for a visual confirmation.