Question

solve the following system of equations graphically on the set of axes below.
$y = \frac{1}{2}x + 6$
$y = -\frac{5}{4}x - 8$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Find intercepts for $y=\frac{1}{2}x+6$

• y-intercept (x=0): $y=\frac{1}{2}(0)+6=6$, so point $(0,6)$
• x-intercept (y=0): $0=\frac{1}{2}x+6 \implies x=-12$, so point $(-12,0)$

Step2: Find intercepts for $y=-\frac{5}{4}x-8$

• y-intercept (x=0): $y=-\frac{5}{4}(0)-8=-8$, so point $(0,-8)$
• x-intercept (y=0): $0=-\frac{5}{4}x-8 \implies x=-\frac{32}{5}=-6.4$, so point $(-6.4,0)$

Step3: Solve algebraically for intersection

Set equations equal:
$$\frac{1}{2}x+6 = -\frac{5}{4}x-8$$
Multiply by 4 to eliminate denominators:
$$2x+24=-5x-32$$
$$2x+5x=-32-24$$
$$7x=-56$$
$$x=-8$$
Substitute $x=-8$ into $y=\frac{1}{2}x+6$:
$$y=\frac{1}{2}(-8)+6=-4+6=2$$

Answer:

The solution (intersection point) is $(-8, 2)$
To plot the lines:

1. For $y=\frac{1}{2}x+6$: plot $(0,6)$ and $(-12,0)$, then draw the line through them.
2. For $y=-\frac{5}{4}x-8$: plot $(0,-8)$ and $(-6.4,0)$, then draw the line through them. The lines cross at $(-8,2)$.