Certainly! Let's solve the given system of equations graphically step by step.
We have the following system of equations:
[tex]\[
\begin{array}{l}
x + y = 0 \\
x - y + 2 = 0
\end{array}
\][/tex]
### Step 1: Rearrange the Equations
First, let's rearrange the equations to express [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex] so we can graph them easily.
1. For the first equation [tex]\( x + y = 0 \)[/tex], we get:
[tex]\[
y = -x
\][/tex]
2. For the second equation [tex]\( x - y + 2 = 0 \)[/tex], we get:
[tex]\[
x - y = -2 \implies -y = -x - 2 \implies y = x + 2
\][/tex]
So, our system now looks like:
[tex]\[
\begin{array}{l}
y = -x \\
y = x + 2
\end{array}
\][/tex]
### Step 2: Plot the Equations
Next, we can plot these two equations on graph paper.
Equation 1: [tex]\( y = -x \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]. So, one point is (0, 0).
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -1 \)[/tex]. So, another point is (1, -1).
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 1 \)[/tex]. Another point is (-1, 1).
Equation 2: [tex]\( y = x + 2 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. So, one point is (0, 2).
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex]. Another point is (1, 3).
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 1 \)[/tex]. Another point is (-1, 1).
### Step 3: Draw the Lines
Using a straight edge, carefully draw the two lines based on the points:
- For [tex]\( y = -x \)[/tex]:
- Connect the points (0, 0), (1, -1), and (-1, 1).
- For [tex]\( y = x + 2 \)[/tex]:
- Connect the points (0, 2), (1, 3), and (-1, 1).
### Step 4: Find the Intersection
The solution to the system of equations is where the two lines intersect.
By looking at the points we noted:
- For [tex]\( y = -x \)[/tex]: the point (-1, 1)
- For [tex]\( y = x + 2 \)[/tex]: the point (-1, 1)
We can see that both lines intersect at the point [tex]\((-1, 1)\)[/tex].
### Solution Set
The solution to the system of equations is:
[tex]\[
(x, y) = (-1, 1)
\][/tex]
So, your final solution set is [tex]\((-1, 1)\)[/tex].
Please make sure to neatly draw the graphs on graph paper, label each equation, and highlight the intersection point for clarity. Then submit your graph paper as instructed.
