Question

solve the following system of equations graphically on the set of axes below.
$y = \frac{3}{2}x - 3$
$x + y = 7$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze the first equation \( y = \frac{3}{2}x - 3 \)

This is in slope - intercept form \( y=mx + b \), where the slope \( m=\frac{3}{2} \) and the y - intercept \( b=-3 \). To graph this line, we can start by plotting the y - intercept. The y - intercept is the point \( (0,-3) \). Then, using the slope, from the point \( (0,-3) \), we move up 3 units and to the right 2 units (since slope \( \frac{\text{rise}}{\text{run}}=\frac{3}{2} \)) to get another point, say \( (2,0) \) (because \( y=\frac{3}{2}(2)-3=3 - 3 = 0 \)).

Step2: Analyze the second equation \( x + y=7 \)

We can rewrite this equation in slope - intercept form \( y=-x + 7 \). Here, the slope \( m=-1 \) and the y - intercept \( b = 7 \). Plot the y - intercept \( (0,7) \). Then, using the slope, from the point \( (0,7) \), we move down 1 unit and to the right 1 unit (since slope \( \frac{\text{rise}}{\text{run}}=\frac{- 1}{1} \)) to get another point, say \( (1,6) \) (because \( y=-1+7 = 6 \)), or we can find the x - intercept by setting \( y = 0 \), then \( x=7 \), so the x - intercept is \( (7,0) \).

Step3: Find the intersection point

When we graph both lines, the point where they intersect is the solution to the system of equations. We can also solve the system algebraically to find the intersection point (to verify). From the second equation \( y=7 - x \). Substitute this into the first equation:
\( 7 - x=\frac{3}{2}x-3 \)
Add \( x \) to both sides: \( 7=\frac{3}{2}x+x - 3=\frac{3x + 2x}{2}-3=\frac{5x}{2}-3 \)
Add 3 to both sides: \( 10=\frac{5x}{2} \)
Multiply both sides by \( \frac{2}{5} \): \( x = 4 \)
Substitute \( x = 4 \) into \( y=7 - x \), we get \( y=7 - 4=3 \)

Answer:

The solution to the system of equations is \( x = 4 \), \( y = 3 \) (the intersection point of the two lines is \( (4,3) \))