Question

solve the following system of equations graphically on the set of axes below. y = 3x - 8 y = -2x - 3 plot two lines by clicking the graph. click a line to delete it.

Explanation:

Step1: Analyze \( y = 3x - 8 \)

For the line \( y = 3x - 8 \), the slope \( m = 3 \) and the y - intercept \( b=-8 \). To plot this line, we can find two points. When \( x = 0 \), \( y=-8 \), so one point is \( (0, - 8) \). When \( y = 0 \), \( 0=3x - 8\), then \( 3x=8\), \( x=\frac{8}{3}\approx2.67 \), so another point is \( (\frac{8}{3},0) \).

Step2: Analyze \( y=-2x - 3 \)

For the line \( y=-2x - 3 \), the slope \( m=-2 \) and the y - intercept \( b = - 3 \). When \( x = 0 \), \( y=-3 \), so one point is \( (0,-3) \). When \( y = 0 \), \( 0=-2x-3\), then \( 2x=-3\), \( x =-\frac{3}{2}=-1.5 \), so another point is \( (-\frac{3}{2},0) \).

Step3: Find the intersection point (solve the system algebraically to confirm)

Set \( 3x - 8=-2x - 3 \)
Add \( 2x \) to both sides: \( 3x+2x-8=-2x + 2x-3\)
\( 5x-8=-3 \)
Add 8 to both sides: \( 5x-8 + 8=-3 + 8\)
\( 5x=5 \)
Divide both sides by 5: \( x = 1 \)
Substitute \( x = 1 \) into \( y = 3x-8 \), \( y=3(1)-8=3 - 8=-5 \)
So the intersection point (solution of the system) is \( (1,-5) \). When we plot the two lines, they will intersect at \( (1,-5) \).

Answer:

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