To graph the system of equations and find their solution, follow these steps:
### Step 1: Express each equation in slope-intercept form (y = mx + b)
1. For the first equation: [tex]\(2x - 3y = -18\)[/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
2x - 3y = -18 \implies -3y = -2x - 18 \implies y = \frac{2}{3}x + 6
\][/tex]
Slope-intercept form: [tex]\(y = \frac{2}{3}x + 6\)[/tex]
2. For the second equation: [tex]\(3x + y = -5\)[/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
3x + y = -5 \implies y = -3x - 5
\][/tex]
Slope-intercept form: [tex]\(y = -3x - 5\)[/tex]
### Step 2: Graph each equation
1. Graph [tex]\(y = \frac{2}{3}x + 6\)[/tex]
- Intercepts:
- y-intercept: When [tex]\(x = 0\)[/tex]:
[tex]\[
y = \frac{2}{3}(0) + 6 = 6 \implies (0, 6)
\][/tex]
- x-intercept: When [tex]\(y = 0\)[/tex]:
[tex]\[
0 = \frac{2}{3}x + 6 \implies \frac{2}{3}x = -6 \implies x = -9 \implies (-9, 0)
\][/tex]
- Points: Plot [tex]\((0, 6)\)[/tex] and [tex]\((-9, 0)\)[/tex] on the graph.
2. Graph [tex]\(y = -3x - 5\)[/tex]
- Intercepts:
- y-intercept: When [tex]\(x = 0\)[/tex]:
[tex]\[
y = -3(0) - 5 = -5 \implies (0, -5)
\][/tex]
- x-intercept: When [tex]\(y = 0\)[/tex]:
[tex]\[
0 = -3x - 5 \implies -3x = 5 \implies x = -\frac{5}{3} \approx -1.67 \implies (-\frac{5}{3}, 0)
\][/tex]
- Points: Plot [tex]\((0, -5)\)[/tex] and [tex]\((-1.67, 0)\)[/tex] on the graph.
### Step 3: Draw the lines
1. Draw the line through the points [tex]\((0, 6)\)[/tex] and [tex]\((-9, 0)\)[/tex] for the first equation.
2. Draw the line through the points [tex]\((0, -5)\)[/tex] and [tex]\((-1.67, 0)\)[/tex] for the second equation.
### Step 4: Find and plot the intersection point
The lines intersect at [tex]\((-3, 4)\)[/tex].
### Conclusion
The solution to the system of equations is:
[tex]\[
(x, y) = (-3, 4)
\][/tex]
Plot the point [tex]\((-3, 4)\)[/tex] on the graph and mark it as the solution.
