Question

solve the following system of equations graphically on the set of axes below.
$y = -dfrac{1}{2}x - 2$
$x - 6y = -12$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Rewrite 2nd equation to slope-intercept

Rearrange $x - 6y = -12$ to $y=mx+b$:
$-6y = -x -12$
$y = \frac{1}{6}x + 2$

Step2: Find points for first line

For $y = -\frac{1}{2}x - 2$:

• When $x=0$, $y=-2$ → $(0, -2)$
• When $x=4$, $y=-\frac{1}{2}(4)-2=-4$ → $(4, -4)$

Step3: Find points for second line

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

• When $x=0$, $y=2$ → $(0, 2)$
• When $x=6$, $y=\frac{1}{6}(6)+2=3$ → $(6, 3)$

Step4: Find intersection algebraically

Set equations equal:
$-\frac{1}{2}x - 2 = \frac{1}{6}x + 2$
Multiply by 6 to eliminate denominators:
$-3x -12 = x + 12$
$-4x = 24$
$x = -6$
Substitute $x=-6$ into $y = -\frac{1}{2}x -2$:
$y = -\frac{1}{2}(-6)-2=3-2=1$

Answer:

The solution (intersection point) is $(-6, 1)$
To plot the lines:

1. For $y = -\frac{1}{2}x - 2$: plot $(0, -2)$ and $(4, -4)$, then draw the line through them.
2. For $x - 6y = -12$ (or $y=\frac{1}{6}x+2$): plot $(0, 2)$ and $(6, 3)$, then draw the line through them. The lines cross at $(-6, 1)$.