Question

solve the following system of equations graphically on the set of axes below.
$y = \frac{1}{5}x - 8$
$y = -\frac{4}{5}x - 3$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze the first equation \( y = \frac{1}{5}x - 8 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m=\frac{1}{5} \) and the y - intercept \( b=-8 \). To graph this line, we can start by plotting the y - intercept. The y - intercept is the point \( (0,-8) \). Then, using the slope \( \frac{1}{5} \) (which means for every 5 units we move to the right along the x - axis, we move up 1 unit along the y - axis), we can find another point. For example, if we move \( x = 5 \) units to the right from \( x = 0 \), then \( y=\frac{1}{5}(5)-8=1 - 8=-7 \). So the point \( (5,-7) \) is also on this line.

Step2: Analyze the second equation \( y=-\frac{4}{5}x - 3 \)

This is also a linear equation in slope - intercept form with slope \( m =-\frac{4}{5} \) and y - intercept \( b=-3 \). We start by plotting the y - intercept \( (0,-3) \). Using the slope \( -\frac{4}{5} \) (for every 5 units we move to the right along the x - axis, we move down 4 units along the y - axis), if we move \( x = 5 \) units to the right from \( x = 0 \), then \( y=-\frac{4}{5}(5)-3=-4 - 3=-7 \). So the point \( (5,-7) \) is on this line.

Step3: Find the intersection point

The solution to the system of linear equations is the point where the two lines intersect. From our calculations above, both lines pass through the point \( (5,-7) \). To confirm, we can set the two equations equal to each other:
\[
\frac{1}{5}x-8=-\frac{4}{5}x - 3
\]
Add \( \frac{4}{5}x \) to both sides:
\[
\frac{1}{5}x+\frac{4}{5}x-8=-\frac{4}{5}x+\frac{4}{5}x - 3
\]
\[
x-8=-3
\]
Add 8 to both sides:
\[
x=-3 + 8
\]
\[
x = 5
\]
Substitute \( x = 5 \) into the first equation \( y=\frac{1}{5}(5)-8=1 - 8=-7 \)

Answer:

The solution to the system of equations is the point \( (5,-7) \)