Question

Graph these equations:
2x–3y=12
x–y=3
Click to select points on the graph.

Answer 1

To graph the equations \(2x - 3y = 12\) and \(x - y = 3\), we first need to convert them into slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.

1. **Equation 1: \(2x - 3y = 12\)**

Let's solve for \(y\):

\[
-3y = -2x + 12
\]
\[
y = \frac{2}{3}x - 4
\]

The slope is \(\frac{2}{3}\) and the y-intercept is \(-4\).

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

Rearranging it gives:

\[
-y = -x + 3
\]
\[
y = x - 3
\]

The slope is \(1\) and the y-intercept is \(-3\).

Now we can identify points for both lines to create the graph.

### Finding Points

**For the first equation \(y = \frac{2}{3}x - 4\):**

1. When \(x = 0\):
\[
y = \frac{2}{3}(0) - 4 = -4 \quad \text{(Point: (0, -4))}
\]

2. When \(x = 6\):
\[
y = \frac{2}{3}(6) - 4 = 2 - 4 = -2 \quad \text{(Point: (6, -2))}
\]

**For the second equation \(y = x - 3\):**

1. When \(x = 0\):
\[
y = 0 - 3 = -3 \quad \text{(Point: (0, -3))}
\]

2. When \(x = 3\):
\[
y = 3 - 3 = 0 \quad \text{(Point: (3, 0))}
\]

### Points Summary
- For \(2x - 3y = 12\): (0, -4), (6, -2)
- For \(x - y = 3\): (0, -3), (3, 0)

### Graphing
Now, you can plot the points on a graph:

- Plot (0, -4) and (6, -2) for the line \(2x - 3y = 12\).
- Plot (0, -3) and (3, 0) for the line \(x - y = 3\).

Draw the lines through the points for each equation and find their intersection, which is the solution of the system of equations.

### Intersection Point
To find the intersection point, you can solve the two equations simultaneously:
- Substitute \(y = x - 3\) into \(2x - 3y = 12\):

\[
2x - 3(x - 3) = 12
\]
\[
2x - 3x + 9 = 12
\]
\[
-x + 9 = 12 \implies x = -3
\]
\[
y = -3 - 3 = -6
\]

Thus, the intersection point is \((-3, -6)\). This gives you both lines and the point where they intersect on your graph.