Question

Graph these equations:
x–3y=9
x=3
Click to select points on the graph.

Answer 1

To graph the given equations, let's start with each one separately.

1. **Equation 1:** \( x - 3y = 9 \)

We can rearrange the equation to the slope-intercept form (y = mx + b):
\[
-3y = -x + 9 \implies y = \frac{1}{3}x - 3
\]
This tells us that the slope (m) is \(\frac{1}{3}\) and the y-intercept (b) is \(-3\).

- Y-intercept: (0, -3)
- To find another point, plug in \(x = 3\):
\[
y = \frac{1}{3}(3) - 3 = 1 - 3 = -2 \quad \Rightarrow \quad (3, -2)
\]
- For \(x = 6\):
\[
y = \frac{1}{3}(6) - 3 = 2 - 3 = -1 \quad \Rightarrow \quad (6, -1)
\]

Now we have points for the line: (0, -3), (3, -2), and (6, -1).

2. **Equation 2:** \( x = 3 \)

This equation represents a vertical line at \(x = 3\). It extends infinitely in the y-direction.

### Graphing the Equations

- **For the line \(x - 3y = 9\)**, plot the points:
- (0, -3)
- (3, -2)
- (6, -1)

- **For the line \(x = 3\)**, draw a vertical dashed line through \(x = 3\).

### Identification of Points
The points where these two equations intersect can also be found. Setting \(x = 3\) in our first equation:
\[
3 - 3y = 9 \implies -3y = 6 \implies y = -2
\]
So the point of intersection is (3, -2).

### Summary of Graph
1. The line for \(x - 3y = 9\) passes through points: (0, -3), (3, -2), (6, -1).
2. The vertical line \(x = 3\) passes through all points of the form (3, y).

You can plot these points and draw the lines to visualize their intersection. If you're using graphing software or graph paper, the intersection point (3, -2) is particularly important!