Question

Graph these equations:
2x+3y=
–
6
x+2y=
–
6
Click to select points on the graph.Graph these equations:
2x+3y=
–
6
x+2y=
–
6
Click to select points on the graph.

Answer 1

To graph the equations \(2x + 3y = -6\) and \(x + 2y = -6\), we first need to rewrite them in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### Equation 1: \(2x + 3y = -6\)

1. Solve for \(y\):
\[
3y = -2x - 6
\]
\[
y = -\frac{2}{3}x - 2
\]

### Equation 2: \(x + 2y = -6\)

1. Solve for \(y\):
\[
2y = -x - 6
\]
\[
y = -\frac{1}{2}x - 3
\]

### Finding Points to Graph

Now we can choose a couple of values for \(x\) to find corresponding values of \(y\) for each equation.

#### For \(2x + 3y = -6\):
- When \(x = 0\):
\[
y = -2 \quad \text{(Point (0, -2))}
\]
- When \(x = -3\):
\[
y = -\frac{2}{3}(-3) - 2 = 2 - 2 = 0 \quad \text{(Point (-3, 0))}
\]

#### For \(x + 2y = -6\):
- When \(x = 0\):
\[
y = -3 \quad \text{(Point (0, -3))}
\]
- When \(x = -6\):
\[
y = -\frac{1}{2}(-6) - 3 = 3 - 3 = 0 \quad \text{(Point (-6, 0))}
\]

### Summary of Points
- For \(2x + 3y = -6\):
- (0, -2)
- (-3, 0)

- For \(x + 2y = -6\):
- (0, -3)
- (-6, 0)

### Graphing The Equations
To graph the equations, plot the points on a Cartesian plane and draw the lines through them:

1. **For \(2x + 3y = -6\)**, plot the points (0, -2) and (-3, 0).
2. **For \(x + 2y = -6\)**, plot the points (0, -3) and (-6, 0).

Then extend the lines through these points. You'll see where they intersect, if at all. The intersection will represent the solution to the system of equations.

Let me know if you need further assistance or a more detailed explanation!