Question

Graph the system of equations to find the solution. You must show ALL work to earn full credit.
y=-x+2
y=3x+6

Answer 1

To graph the system of equations \( y = -x + 2 \) and \( y = 3x + 6 \), we'll first rewrite each equation in slope-intercept form, which they already are, and then plot the lines.

### 1. Determine Points for Each Line

#### For the first equation: \( y = -x + 2 \)

- **y-intercept**: Set \( x = 0 \)
\[
y = -0 + 2 = 2 \quad \text{(Point: (0, 2))}
\]
- **x-intercept**: Set \( y = 0 \)
\[
0 = -x + 2 \implies x = 2 \quad \text{(Point: (2, 0))}
\]

So, the first line passes through the points \( (0, 2) \) and \( (2, 0) \).

#### For the second equation: \( y = 3x + 6 \)

- **y-intercept**: Set \( x = 0 \)
\[
y = 3(0) + 6 = 6 \quad \text{(Point: (0, 6))}
\]
- **x-intercept**: Set \( y = 0 \)
\[
0 = 3x + 6 \implies 3x = -6 \implies x = -2 \quad \text{(Point: (-2, 0))}
\]

So, the second line passes through the points \( (0, 6) \) and \( (-2, 0) \).

### 2. Graph the Equations

1. **Graph \( y = -x + 2 \)**:
- Plot points \( (0, 2) \) and \( (2, 0) \).
- Draw the line through these points. The line will slope downward from left to right.

2. **Graph \( y = 3x + 6 \)**:
- Plot points \( (0, 6) \) and \( (-2, 0) \).
- Draw the line through these points. The line will slope upward from left to right.

### 3. Find the Intersection Point

To find the exact point of intersection, we can set the two equations equal to each other:
\[
-x + 2 = 3x + 6
\]

### Solving for \( x \):
1. Add \( x \) to both sides:
\[
2 = 4x + 6
\]
2. Subtract 6 from both sides:
\[
-4 = 4x
\]
3. Divide by 4:
\[
x = -1
\]

### Find \( y \) using \( x = -1 \):

Substituting \( x = -1 \) into either original equation (we'll use \( y = -x + 2 \)):
\[
y = -(-1) + 2 = 1 + 2 = 3
\]

### Conclusion

The solution to the system of equations is:
\[
\boxed{(-1, 3)}
\]

### Final Note

In the graph, you would see that both lines intersect at the point \( (-1, 3) \), which confirms our algebraic solution.