Question

Solve the following system of equations graphically on the set of axes below. y=3/2x-3 x+y=7

Answer 1

To solve the system of equations graphically, we need to plot both equations on the same set of axes.

1. **First Equation: \( y = \frac{3}{2}x - 3 \)**
- This is in slope-intercept form \( y = mx + b \), where \( m = \frac{3}{2} \) is the slope and \( b = -3 \) is the y-intercept.
- To find points for this line, we can calculate the y-value for several x-values:

- If \( x = 0 \):
\[
y = \frac{3}{2}(0) - 3 = -3 \quad \text{(point: (0, -3))}
\]
- If \( x = 2 \):
\[
y = \frac{3}{2}(2) - 3 = 3 - 3 = 0 \quad \text{(point: (2, 0))}
\]
- If \( x = 4 \):
\[
y = \frac{3}{2}(4) - 3 = 6 - 3 = 3 \quad \text{(point: (4, 3))}
\]

- Plot these points (0, -3), (2, 0), and (4, 3) and draw the line.

2. **Second Equation: \( x + y = 7 \)**
- Rearranging this into slope-intercept form gives \( y = -x + 7 \).
- To find points for this line, we can calculate the y-value for several x-values:

- If \( x = 0 \):
\[
y = -0 + 7 = 7 \quad \text{(point: (0, 7))}
\]
- If \( x = 7 \):
\[
y = -7 + 7 = 0 \quad \text{(point: (7, 0))}
\]
- If \( x = 3 \):
\[
y = -3 + 7 = 4 \quad \text{(point: (3, 4))}
\]

- Plot these points (0, 7), (7, 0), and (3, 4) and draw the line.

3. **Finding the Intersection Point:**
- The solution to the system of equations is where the two lines intersect.
- By visually inspecting the plotted lines, we can find the coordinates of the intersection point.

### Intersection Calculation:
To find the exact intersection algebraically, we can substitute \( y \) from the first equation into the second equation:

Substitute \( y = \frac{3}{2}x - 3 \) into \( x + y = 7 \):

\[
x + \left(\frac{3}{2}x - 3\right) = 7
\]
\[
x + \frac{3}{2}x - 3 = 7
\]
\[
\frac{5}{2}x - 3 = 7
\]
Add 3 to both sides:
\[
\frac{5}{2}x = 10
\]
Multiply both sides by \(\frac{2}{5}\):
\[
x = 4
\]

Now, substitute \( x = 4 \) back into the first equation to find \( y \):
\[
y = \frac{3}{2}(4) - 3 = 6 - 3 = 3
\]

### Solution:
The solution to the system of equations is \( (4, 3) \). This point is where the two lines intersect on the graph.