To solve the equations \(3x - 2y = 4\) and \(5x - 2y = 0\) graphically, we will follow these steps:
### Step 1: Rewrite the equations in slope-intercept form
We will convert both equations into the form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
1. For the first equation \(3x - 2y = 4\):
\[
-2y = -3x + 4 \implies y = \frac{3}{2}x - 2
\]
2. For the second equation \(5x - 2y = 0\):
\[
-2y = -5x \implies y = \frac{5}{2}x
\]
### Step 2: Find points for the first equation
To plot the first line, we will find two points by substituting values for \(x\).
1. Let \(x = 0\):
\[
y = \frac{3}{2}(0) - 2 = -2 \quad \text{(Point: (0, -2))}
\]
2. Let \(y = 7\):
\[
7 = \frac{3}{2}x - 2 \implies \frac{3}{2}x = 9 \implies x = 6 \quad \text{(Point: (6, 7))}
\]
### Step 3: Find points for the second equation
To plot the second line, we will find two points by substituting values for \(x\).
1. Let \(x = 0\):
\[
y = \frac{5}{2}(0) = 0 \quad \text{(Point: (0, 0))}
\]
2. Let \(y = 5\):
\[
5 = \frac{5}{2}x \implies x = 2 \quad \text{(Point: (2, 5))}
\]
### Step 4: Plot the points on a graph
- For the first equation \(3x - 2y = 4\), plot the points \((0, -2)\) and \((6, 7)\).
- For the second equation \(5x - 2y = 0\), plot the points \((0, 0)\) and \((2, 5)\).
### Step 5: Draw the lines
- Draw a line through the points \((0, -2)\) and \((6, 7)\).
- Draw a line through the points \((0, 0)\) and \((2, 5)\).
### Step 6: Analyze the graph
- Upon drawing the two lines, observe their slopes. The first line has a slope of \(\frac{3}{2}\) and the second line has a slope of \(\frac{5}{2}\).
- Since the slopes are different, the lines will intersect at a point.
### Step 7: Find the point of intersection
To find the intersection point, we can set the two equations equal to each other:
\[
\frac{3}{2}x - 2 = \frac{5}{2}x
\]
Solving for \(x\):
\[
-2 = \frac{5}{2}x - \frac{3}{2}x \implies -2 = \frac{2}{2}x \implies x = -2
\]
Substituting \(x = -2\) back into either equation to find \(y\):
\[
y = \frac{5}{2}(-2) = -5
\]
### Final Solution
The solution to the system of equations is the point of intersection:
\[
\text{Point of intersection: } (-2, -5)
\]
