Question

solve the following system of equations graphically on the set of axes below.
$y = x + 6$
$y = -dfrac{3}{2}x - 4$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze \( y = x + 6 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = 1 \) and the y - intercept \( b = 6 \). To graph this line, we can start by plotting the y - intercept. The y - intercept is the point where \( x = 0 \), so when \( x = 0 \), \( y=0 + 6=6 \). So we have the point \( (0,6) \). Then, using the slope (rise over run), since the slope is 1 (which is \( \frac{1}{1} \)), from the point \( (0,6) \), we can move 1 unit up (rise) and 1 unit to the right (run) to get the next point \( (1,7) \), or 1 unit down and 1 unit to the left to get \( (- 1,5) \).

Step2: Analyze \( y=-\frac{3}{2}x - 4 \)

This is also in slope - intercept form \( y = mx + b \), where the slope \( m=-\frac{3}{2} \) and the y - intercept \( b=-4 \). The y - intercept is the point where \( x = 0 \), so when \( x = 0 \), \( y=0-4=-4 \). So we have the point \( (0, - 4) \). The slope is \( -\frac{3}{2} \), which means for every 2 units we move to the right (run), we move 3 units down (rise) (because the slope is negative). From the point \( (0,-4) \), if we move 2 units to the right (to \( x = 2 \)) and 3 units down, we get \( y=-4-3=-7 \), so the point \( (2,-7) \). Or, if we move 2 units to the left (to \( x=-2 \)) and 3 units up, we get \( y=-4 + 3=-1 \), so the point \( (-2,-1) \).

Step3: Find the intersection point

To solve the system of equations graphically, we need to find the point where the two lines intersect. We can also solve the system algebraically to confirm. Set the two equations equal to each other:
\( x + 6=-\frac{3}{2}x-4 \)
Add \( \frac{3}{2}x \) to both sides:
\( x+\frac{3}{2}x+6=-\frac{3}{2}x+\frac{3}{2}x - 4 \)
\( \frac{2x + 3x}{2}+6=-4 \)
\( \frac{5x}{2}+6=-4 \)
Subtract 6 from both sides:
\( \frac{5x}{2}+6 - 6=-4 - 6 \)
\( \frac{5x}{2}=-10 \)
Multiply both sides by \( \frac{2}{5} \):
\( x=-10\times\frac{2}{5}=-4 \)
Now substitute \( x = - 4 \) into \( y=x + 6 \):
\( y=-4 + 6=2 \)

Answer:

The solution to the system of equations is the point of intersection of the two lines, which is \( (-4,2) \)