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To solve the given system of linear equations by graphing, we first need to rewrite both equations in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

The system is:

1. \( 3x - 2y = 0 \)
2. \( 9x - 2y = 12 \)

### Step 1: Solve the first equation for \( y \)
\[
3x - 2y = 0
\]
Subtract \( 3x \) from both sides:
\[
-2y = -3x
\]
Now, divide by \( -2 \):
\[
y = \frac{3}{2}x
\]
The slope of the first line is \( \frac{3}{2} \), and the y-intercept is \( 0 \).

### Step 2: Solve the second equation for \( y \)
\[
9x - 2y = 12
\]
Subtract \( 9x \) from both sides:
\[
-2y = -9x + 12
\]
Now, divide by \( -2 \):
\[
y = \frac{9}{2}x - 6
\]
The slope of the second line is \( \frac{9}{2} \), and the y-intercept is \( -6 \).

### Step 3: Graph the system

- For the first line \( y = \frac{3}{2}x \), start at \( (0, 0) \), and use the slope \( \frac{3}{2} \) to find the next point. From \( (0, 0) \), move 2 units right and 3 units up to \( (2, 3) \).
- For the second line \( y = \frac{9}{2}x - 6 \), start at \( (0, -6) \), and use the slope \( \frac{9}{2} \). From \( (0, -6) \), move 2 units right and 9 units up to \( (2, 3) \).

### Step 4: Find the intersection
Both lines intersect at the point \( (2, 3) \). Therefore, the solution to the system of equations is \( (2, 3) \).

Would you like a more detailed explanation or further assistance?

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Here are 5 related questions to expand your understanding:

1. How do you determine the slope from the standard form of a line?
2. What does it mean for two lines to intersect at a point?
3. Can two lines with the same slope have different y-intercepts? What does that imply?
4. How would the graph change if the second equation were \( 9x - 2y = -12 \)?
5. What methods can be used to solve a system of equations besides graphing?

**Tip**: Always check if the lines intersect by setting their equations equal to each other!

Solve the system by graphing: 1) 3x - 2y = 0 2) 9x - 2y = 12

Learn how to solve the system of linear equations 3x - 2y = 0 and 9x - 2y = 12 by graphing. This solution demonstrates converting equations to slope-intercept form, identifying slopes and intercepts, and determining the intersection point of the two lines. Suitable for students in grades 9-11.

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