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 $y=\frac{1}{2}x+6$

• x-intercept: Set $y=0$:

$0=\frac{1}{2}x+6 \implies \frac{1}{2}x=-6 \implies x=-12$

• y-intercept: Set $x=0$:

$y=\frac{1}{2}(0)+6=6$
Points: $(-12, 0)$ and $(0, 6)$

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

• x-intercept: Set $y=0$:

$0=-\frac{5}{4}x-8 \implies \frac{5}{4}x=-8 \implies x=-\frac{32}{5}=-6.4$

• y-intercept: Set $x=0$:

$y=-\frac{5}{4}(0)-8=-8$
Points: $(-6.4, 0)$ and $(0, -8)$

Step3: Solve algebraically for intersection

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

Answer:

The solution (intersection point) of the system is $\boldsymbol{(-8, 2)}$
To graph:

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

The two lines cross at $(-8, 2)$.