To solve the given system of equations graphically, follow these steps:
### Step 1: Rewrite the equations in slope-intercept form if needed.
1. First equation: [tex]\( y = -\frac{1}{2} x - 8 \)[/tex]
- This equation is already in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where [tex]\( m = -\frac{1}{2} \)[/tex] and [tex]\( b = -8 \)[/tex].
2. Second equation: [tex]\( x - y = 2 \)[/tex]
- Rearrange to get [tex]\( y \)[/tex] on one side:
[tex]\[
x - y = 2 \implies y = x - 2
\][/tex]
- Now this is also in slope-intercept form, where [tex]\( m = 1 \)[/tex] and [tex]\( b = -2 \)[/tex].
### Step 2: Plot the first equation [tex]\( y = -\frac{1}{2} x - 8 \)[/tex].
- Identify the y-intercept (where [tex]\( x=0 \)[/tex]):
[tex]\[
y = -8
\][/tex]
So, the point is (0, -8).
- Calculate another point using the slope [tex]\( m = -\frac{1}{2} \)[/tex]:
If [tex]\( x = 2 \)[/tex], then:
[tex]\[
y = -\frac{1}{2} \times 2 - 8 = -1 - 8 = -9
\][/tex]
So, another point is (2, -9).
- Plot these points and draw the line through them.
### Step 3: Plot the second equation [tex]\( y = x - 2 \)[/tex].
- Identify the y-intercept (where [tex]\( x=0 \)[/tex]):
[tex]\[
y = -2
\][/tex]
So, the point is (0, -2).
- Calculate another point using the slope [tex]\( m = 1 \)[/tex]:
If [tex]\( x = 2 \)[/tex], then:
[tex]\[
y = 2 - 2 = 0
\][/tex]
So, another point is (2, 0).
- Plot these points and draw the line through them.
### Step 4: Find the intersection of the two lines.
- Graph both lines on the same axes.
- The point where the two lines intersect is the solution to the system of equations.
After plotting, you will observe that the lines intersect at the point [tex]\((-4, -6)\)[/tex].
### Solution:
[tex]\[
\boxed{(-4, -6)}
\][/tex]
The point [tex]\((-4, -6)\)[/tex] is the solution to the system of equations, meaning [tex]\( x = -4 \)[/tex] and [tex]\( y = -6 \)[/tex] satisfy both equations simultaneously.
