To solve the system of equations by graphing, we need to graph each of the lines and find their intersection point.
The system of equations given is:
[tex]\[
\left\{
\begin{array}{l}
y = -x - 3 \\
y = -4x - 3
\end{array}
\right.
\][/tex]
### Step-by-Step Solution
1. Graph the first equation [tex]\( y = -x - 3 \)[/tex]:
- Identify two points on the line by choosing values for [tex]\( x \)[/tex] and solving for [tex]\( y \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -0 - 3 = -3
\][/tex]
Thus, one point is [tex]\( (0, -3) \)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[
y = -3 - 3 = -6
\][/tex]
Thus, another point is [tex]\( (3, -6) \)[/tex].
- Plot these points and draw the line through them.
2. Graph the second equation [tex]\( y = -4x - 3 \)[/tex]:
- Similarly, choose two values for [tex]\( x \)[/tex] and solve for [tex]\( y \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -4(0) - 3 = -3
\][/tex]
Thus, one point is [tex]\( (0, -3) \)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = -4(1) - 3 = -7
\][/tex]
Thus, another point is [tex]\( (1, -7) \)[/tex].
- Plot these points and draw the line through them.
3. Find the intersection point:
- By solving the equations simultaneously (setting [tex]\( -x - 3 = -4x - 3 \)[/tex]), we get:
[tex]\[
-x - 3 = -4x - 3
\][/tex]
Simplifying, we get:
[tex]\[
-x + 4x = 0 \\
3x = 0 \\
x = 0
\][/tex]
- Substitute [tex]\( x = 0 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[
y = -0 - 3 = -3
\][/tex]
- The intersection point is [tex]\( (0, -3) \)[/tex].
4. Summary:
- The line [tex]\( y = -x - 3 \)[/tex] passes through the points [tex]\((0, -3)\)[/tex] and [tex]\((3, -6)\)[/tex].
- The line [tex]\( y = -4x - 3 \)[/tex] passes through the points [tex]\((0, -3)\)[/tex] and [tex]\((1, -7)\)[/tex].
- Both lines intersect at the point [tex]\((0, -3)\)[/tex].
[tex]\[
\boxed{(0, -3)}
\][/tex]
By plotting these lines on a graph and visually identifying the intersection point, we confirm that the solution to the system of equations is [tex]\((0, -3)\)[/tex].
