Question

solve the following system of equations graphically on the set of axes below.
$y = x + 8$
$y = -2x - 4$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze \( y = x + 8 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = 1 \) and the y - intercept \( b = 8 \). To graph this line, we can start by plotting the y - intercept. The y - intercept is the point where \( x = 0 \), so when \( x = 0 \), \( y=0 + 8=8 \). So we plot the point \( (0,8) \). Then, using the slope (rise over run), since the slope is 1 (which is \( \frac{1}{1} \)), from the point \( (0,8) \), we can move 1 unit up (rise) and 1 unit to the right (run) to get the next point. For example, when we move from \( (0,8) \) with a rise of 1 and run of 1, we get the point \( (1,9) \). We can also move in the opposite direction: from \( (0,8) \), move 1 unit down (rise of - 1) and 1 unit to the left (run of - 1) to get the point \( (- 1,7) \).

Step2: Analyze \( y=-2x - 4 \)

This is also a linear equation in slope - intercept form \( y = mx + b \), where the slope \( m=-2 \) (or \( \frac{-2}{1} \)) and the y - intercept \( b=-4 \). To graph this line, we first plot the y - intercept. When \( x = 0 \), \( y=-2(0)-4=-4 \), so we plot the point \( (0, - 4) \). Using the slope, since the slope is - 2 (rise of - 2 and run of 1), from the point \( (0,-4) \), we can move 2 units down (rise) and 1 unit to the right (run) to get the next point. For example, moving from \( (0,-4) \) with a rise of - 2 and run of 1, we get the point \( (1,-6) \). We can also move in the opposite direction: from \( (0,-4) \), move 2 units up (rise of 2) and 1 unit to the left (run of - 1) to get the point \( (-1,-2) \).

Step3: Find the intersection point (solution of the system)

The solution of a system of linear equations graphically is the point where the two lines intersect. To find the intersection algebraically (which will help us confirm the graphical solution), we set the two equations equal to each other since at the intersection point, the \( y \) - values (and \( x \) - values) of the two equations are equal.
Set \( x + 8=-2x-4 \)
Add \( 2x \) to both sides of the equation:
\( x+2x + 8=-2x + 2x-4 \)
\( 3x+8=-4 \)
Subtract 8 from both sides:
\( 3x+8 - 8=-4 - 8 \)
\( 3x=-12 \)
Divide both sides by 3:
\( x=\frac{-12}{3}=-4 \)
Now substitute \( x = - 4 \) into one of the original equations, say \( y=x + 8 \). Then \( y=-4 + 8 = 4 \).
So the two lines intersect at the point \( (-4,4) \). When we graph the two lines, the point where they cross each other is \( (-4,4) \).

Answer:

The solution to the system of equations is the point of intersection of the two lines, which is \( x=-4,y = 4 \) or the ordered pair \( (-4,4) \).