To solve the problem step by step, we will first rewrite the equation in a more manageable form, plot the line on a graph, and then find the values of \( m \) and \( n \) for the given points.
### Step 1: Rewrite the equation
The given equation is:
\[
2x - 3y + 12 = 0
\]
We can rearrange this equation to express \( y \) in terms of \( x \):
\[
3y = 2x + 12
\]
\[
y = \frac{2}{3}x + 4
\]
### Step 2: Find points to plot the line
To plot the line, we can choose different values for \( x \) and calculate the corresponding \( y \) values.
1. **For \( x = 0 \)**:
\[
y = \frac{2}{3}(0) + 4 = 4 \quad \Rightarrow \quad (0, 4)
\]
2. **For \( x = 3 \)**:
\[
y = \frac{2}{3}(3) + 4 = 2 + 4 = 6 \quad \Rightarrow \quad (3, 6)
\]
3. **For \( x = -6 \)**:
\[
y = \frac{2}{3}(-6) + 4 = -4 + 4 = 0 \quad \Rightarrow \quad (-6, 0)
\]
### Step 3: Plot the points on graph paper
Using the points calculated, we will plot the points \( (0, 4) \), \( (3, 6) \), and \( (-6, 0) \) on the graph paper. After plotting these points, we will draw a straight line through them.
### Step 4: Find \( m \) when \( y = -2 \)
To find the value of \( m \) such that the point \( (m, -2) \) lies on the line, we substitute \( y = -2 \) into the equation:
\[
-2 = \frac{2}{3}x + 4
\]
Rearranging gives:
\[
-2 - 4 = \frac{2}{3}x
\]
\[
-6 = \frac{2}{3}x
\]
Multiplying both sides by \( \frac{3}{2} \):
\[
x = -6 \cdot \frac{3}{2} = -9
\]
Thus, \( m = -9 \).
### Step 5: Find \( n \) when \( x = 3 \)
Now we find the value of \( n \) such that the point \( (3, n) \) lies on the line. We substitute \( x = 3 \) into the equation:
\[
y = \frac{2}{3}(3) + 4
\]
Calculating gives:
\[
y = 2 + 4 = 6
\]
Thus, \( n = 6 \).
### Final Answer
The values are:
\[
m = -9 \quad \text{and} \quad n = 6
\]
