To solve the system of equations by graphing, we need to graph both equations on the same coordinate plane and identify their point of intersection, which will give us the solution.
The given system of equations is:
[tex]\[
\left\{
\begin{array}{l}
x + y = -3 \\
y = 2x + 9
\end{array}
\right.
\][/tex]
### Step-by-Step Solution:
1. Graph the first equation [tex]\( x + y = -3 \)[/tex]:
- Rewrite the equation in slope-intercept form, [tex]\( y = mx + b \)[/tex]:
[tex]\[
x + y = -3 \implies y = -x - 3
\][/tex]
- Find the y-intercept ([tex]\(b\)[/tex]) and the slope ([tex]\(m\)[/tex]):
- Y-intercept ([tex]\(b\)[/tex]): -3 (point is (0, -3))
- Slope ([tex]\(m\)[/tex]): -1
- Plot the y-intercept (0, -3) on the graph.
- Use the slope to find another point. Since the slope is -1, from (0, -3), move 1 unit right and 1 unit down to get the point (1, -4).
- Draw the line passing through these points.
2. Graph the second equation [tex]\( y = 2x + 9 \)[/tex]:
- The equation is already in slope-intercept form [tex]\( y = mx + b \)[/tex].
- Find the y-intercept ([tex]\(b\)[/tex]) and the slope ([tex]\(m\)[/tex]):
- Y-intercept ([tex]\(b\)[/tex]): 9 (point is (0, 9))
- Slope ([tex]\(m\)[/tex]): 2
- Plot the y-intercept (0, 9) on the graph.
- Use the slope to find another point. Since the slope is 2, from (0, 9), move 1 unit right and 2 units up to get the point (1, 11).
- Draw the line passing through these points.
3. Find the intersection point:
- The intersection point of these two lines is the solution to the system of equations.
- By graphing, you will observe that the intersection point is:
[tex]\[
\boxed{(-4, 1)}
\][/tex]
### Conclusion:
The solution to the system of equations [tex]\( x + y = -3 \)[/tex] and [tex]\( y = 2x + 9 \)[/tex] is the point [tex]\( (-4, 1) \)[/tex]. This means [tex]\( x = -4 \)[/tex] and [tex]\( y = 1 \)[/tex]. This is confirmed by the intersection point found on the graph.
