Question

solve the following system of equations graphically on the set of axes below.
$y = -x + 2$
$y = \frac{3}{5}x - 6$

Explanation:

Step1: Analyze the first equation \( y = -x + 2 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m=- 1 \) and the y - intercept \( b = 2 \). To graph this line, we can start by plotting the y - intercept at the point \( (0,2) \). Then, using the slope (which is \( \frac{\text{rise}}{\text{run}}=\frac{-1}{1} \)), from the point \( (0,2) \), we can move down 1 unit and right 1 unit to get the next point \( (1,1) \), or up 1 unit and left 1 unit to get the point \( (- 1,3) \). We can draw a straight line through these points.

Step2: Analyze the second equation \( y=\frac{3}{5}x-6 \)

This is also a linear equation in slope - intercept form with slope \( m = \frac{3}{5} \) and y - intercept \( b=-6 \). First, plot the y - intercept at the point \( (0,-6) \). Then, using the slope \( \frac{\text{rise}}{\text{run}}=\frac{3}{5} \), from the point \( (0,-6) \), we move up 3 units and right 5 units to get the point \( (5,-3) \), or down 3 units and left 5 units to get the point \( (-5,-9) \). Draw a straight line through these points.

Step3: Find the intersection point

The solution to a system of linear equations graphed is the point where the two lines intersect. By graphing the two lines (either by hand or using a graphing utility), we can see that the two lines intersect at the point \( (5,-3) \). We can verify this algebraically:
Set \( -x + 2=\frac{3}{5}x-6 \)
Add \( x \) to both sides: \( 2=\frac{3}{5}x+x - 6 \)
Combine like terms: \( 2=\frac{3x + 5x}{5}-6=\frac{8x}{5}-6 \)
Add 6 to both sides: \( 2 + 6=\frac{8x}{5}\)
\( 8=\frac{8x}{5}\)
Multiply both sides by \( \frac{5}{8} \): \( x = 5 \)
Substitute \( x = 5 \) into the first equation \( y=-x + 2 \), we get \( y=-5 + 2=-3 \)

Answer:

The solution to the system of equations is \( x = 5,y=-3 \) or the ordered pair \( (5,-3) \)