Question

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

Explanation:

Step1: Analyze \( y = x - 4 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = 1 \) and the y - intercept \( b=-4 \). To graph this line, we can start by plotting the y - intercept at the point \( (0,-4) \). Then, using the slope (rise over run, since \( m = 1=\frac{1}{1} \)), we can find another point. From \( (0,-4) \), we move 1 unit up and 1 unit to the right to get the point \( (1,-3) \), or 1 unit down and 1 unit to the left to get the point \( (-1,-5) \).

Step2: Rewrite \( 3x + y=-8 \) in slope - intercept form

We solve for \( y \):
\( y=-3x - 8 \)
Here, the slope \( m=-3 \) and the y - intercept \( b = - 8 \). To graph this line, we plot the y - intercept at the point \( (0,-8) \). Using the slope \( m=-3=\frac{-3}{1} \), from \( (0,-8) \), we move 3 units down and 1 unit to the right to get the point \( (1,-11) \), or 3 units up and 1 unit to the left to get the point \( (-1,-5) \).

Step3: Find the intersection point

The intersection point of the two lines is the solution to the system of equations. We can also solve the system algebraically to find the intersection point. Set the two equations equal to each other:
\( x - 4=-3x - 8 \)
Add \( 3x \) to both sides:
\( x+3x - 4=-3x + 3x-8 \)
\( 4x-4=-8 \)
Add 4 to both sides:
\( 4x-4 + 4=-8 + 4 \)
\( 4x=-4 \)
Divide both sides by 4:
\( x=-1 \)
Substitute \( x = - 1 \) into \( y=x - 4 \):
\( y=-1-4=-5 \)

Answer:

The solution to the system of equations is \( x=-1,y = - 5 \), so the intersection point (the solution) is \( (-1,-5) \)