Question
Graph these equations:
2x+3y=6
x=
–
6
Click to select points on the graph.
2x+3y=6
x=
–
6
Click to select points on the graph.
Answers
GPT-4o mini
To graph the equations \(2x + 3y = 6\) and \(x = -6\), we'll first convert the linear equation into slope-intercept form, and then we'll plot the two equations on the coordinate plane.
### Step 1: Graph the equation \(2x + 3y = 6\)
1. **Rearrange the equation** to solve for \(y\):
\[
3y = -2x + 6
\]
\[
y = -\frac{2}{3}x + 2
\]
2. **Identify the slope and y-intercept**:
- The slope (\(m\)) is \(-\frac{2}{3}\).
- The y-intercept (\(b\)) is \(2\) (this is the point (0, 2) on the y-axis).
3. **Plot the y-intercept**:
- Start at (0, 2) on the graph.
4. **Use the slope to find another point**:
- From (0, 2), move down 2 units and to the right 3 units to mark the point (3, 0).
- Another point can be found by moving up 2 units and to the left 3 units to get (-3, 4).
5. **Draw the line** through the points(0,2) and (3,0):
### Step 2: Graph the equation \(x = -6\)
1. **Understand what this equation represents**:
- The equation \(x = -6\) is a vertical line that intersects the x-axis at -6.
2. **Plot the vertical line**:
- This line will pass through all points where the x-coordinate is -6.
### Step 3: Combine the graphs
- Draw the horizontal line for \(2x + 3y = 6\) passing through the points you plotted.
- Draw the vertical line for \(x = -6\) straight up and down through the x-coordinate.
### Conclusion
You should have obtained:
1. A sloping line going downwards from left to right that intersects the y-axis at (0, 2) and has points like (3, 0) and (-3, 4).
2. A vertical line crossing at -6 along the x-axis.
If you have graphing software or graph paper, you can easily reflect the points arranged as described above to create a visual representation of both lines.
### Step 1: Graph the equation \(2x + 3y = 6\)
1. **Rearrange the equation** to solve for \(y\):
\[
3y = -2x + 6
\]
\[
y = -\frac{2}{3}x + 2
\]
2. **Identify the slope and y-intercept**:
- The slope (\(m\)) is \(-\frac{2}{3}\).
- The y-intercept (\(b\)) is \(2\) (this is the point (0, 2) on the y-axis).
3. **Plot the y-intercept**:
- Start at (0, 2) on the graph.
4. **Use the slope to find another point**:
- From (0, 2), move down 2 units and to the right 3 units to mark the point (3, 0).
- Another point can be found by moving up 2 units and to the left 3 units to get (-3, 4).
5. **Draw the line** through the points(0,2) and (3,0):
### Step 2: Graph the equation \(x = -6\)
1. **Understand what this equation represents**:
- The equation \(x = -6\) is a vertical line that intersects the x-axis at -6.
2. **Plot the vertical line**:
- This line will pass through all points where the x-coordinate is -6.
### Step 3: Combine the graphs
- Draw the horizontal line for \(2x + 3y = 6\) passing through the points you plotted.
- Draw the vertical line for \(x = -6\) straight up and down through the x-coordinate.
### Conclusion
You should have obtained:
1. A sloping line going downwards from left to right that intersects the y-axis at (0, 2) and has points like (3, 0) and (-3, 4).
2. A vertical line crossing at -6 along the x-axis.
If you have graphing software or graph paper, you can easily reflect the points arranged as described above to create a visual representation of both lines.