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 the first equation \( y = x - 4 \)

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: Analyze the second equation \( 3x + y=-8 \)

We can rewrite this equation in slope - intercept form (\( y=mx + b \)) by solving for \( y \). Subtract \( 3x \) from both sides: \( y=-3x - 8 \). Here, the slope \( m=-3 \) and the y - intercept \( b = - 8 \). To graph this line, we start by plotting the y - intercept at the point \( (0,-8) \). Then, 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 \( (1,-11) \) (but this is outside our visible grid for now), or 3 units up and 1 unit to the left to get the point \( (-1,-5) \).

Step3: Find the intersection point

When we graph both lines, we look for the point where they intersect. From our earlier calculations, we can see that the point \( (-1,-5) \) lies on both lines? Wait, no. Wait, let's solve the system algebraically to check. We have the system:
\[

$$\begin{cases} y=x - 4\\ 3x + y=-8 \end{cases}$$

\]
Substitute \( y=x - 4 \) into the second equation: \( 3x+(x - 4)=-8 \).
Simplify: \( 3x+x-4=-8 \), \( 4x-4=-8 \).
Add 4 to both sides: \( 4x=-8 + 4=-4 \).
Divide both sides by 4: \( x=-1 \).
Now substitute \( x = - 1 \) into \( y=x - 4 \): \( y=-1-4=-5 \).

So the two lines intersect at the point \( (-1,-5) \). When graphing, the line \( y=x - 4 \) passes through \( (0,-4) \), \( (1,-3) \), \( (-1,-5) \) and the line \( y=-3x - 8 \) passes through \( (0,-8) \), \( (-1,-5) \). So the intersection point is \( (-1,-5) \).

Answer:

The solution to the system of equations is \( x=-1 \), \( y = - 5 \) (or the ordered pair \( (-1,-5) \)).