Let's solve the system of equations graphically and identify their point of intersection, which represents the solution to the system.
The system of equations is:
[tex]\[
\begin{array}{l}
y = -x + 2 \\
x - 3y = -18
\end{array}
\][/tex]
### Step-by-Step Graphing Instructions
1. Equation 1: [tex]\(y = -x + 2\)[/tex]
- This is a linear equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
- The slope [tex]\(m = -1\)[/tex] and the y-intercept [tex]\(b = 2\)[/tex].
- Plot the y-intercept at point [tex]\((0, 2)\)[/tex].
- From [tex]\((0, 2)\)[/tex], use the slope to find another point. Since the slope is [tex]\(-1\)[/tex], go down 1 unit and right 1 unit to point [tex]\((1, 1)\)[/tex].
- Draw the line through these points.
2. Equation 2: [tex]\(x - 3y = -18\)[/tex]
- This needs to be converted to slope-intercept form [tex]\(y = mx + b\)[/tex].
- Solve for [tex]\(y\)[/tex] as follows:
[tex]\[
x - 3y = -18 \implies -3y = -x - 18 \implies y = \frac{1}{3}x + 6
\][/tex]
- The slope is [tex]\(\frac{1}{3}\)[/tex] and the y-intercept is [tex]\(6\)[/tex].
- Plot the y-intercept at point [tex]\((0, 6)\)[/tex].
- From [tex]\((0, 6)\)[/tex], use the slope to find another point. Since the slope is [tex]\(\frac{1}{3}\)[/tex], go up 1 unit and right 3 units to point [tex]\((3, 7)\)[/tex].
- Draw the line through these points.
3. Intersection Point
- The solution to the system is where the two lines intersect.
- Upon plotting both lines on the same coordinate plane, identify their intersection point.
From the solution provided, the intersection occurs at [tex]\((-3, 5)\)[/tex]. This point should be clearly marked on the graph.
### Summary
- For the line [tex]\(y = -x + 2\)[/tex], plot points [tex]\((0, 2)\)[/tex] and [tex]\((1, 1)\)[/tex].
- For the line [tex]\(y = \frac{1}{3}x + 6\)[/tex], plot points [tex]\((0, 6)\)[/tex] and [tex]\((3, 7)\)[/tex].
- Mark the intersection point at [tex]\((-3, 5)\)[/tex].
This intersection represents the solution to the system of equations, which is [tex]\((-3, 5)\)[/tex].
Now you can use the provided drawing and marking tools to graph these lines and indicate the solution [tex]\((-3, 5)\)[/tex] on your coordinate plane.
