Question

part 1 of 2
graph each equation. determine the sc
$x + 2y = 12$
$7x + 2y = 0$
use the graphing tool to graph the syst
click to enlarge graph
click the graph, choose a tool in the palette and fol

Explanation:

Step1: Rewrite equations to slope-intercept form

For $x + 2y = 12$:
$2y = -x + 12$
$y = -\frac{1}{2}x + 6$

For $7x + 2y = 0$:
$2y = -7x$
$y = -\frac{7}{2}x$

Step2: Find intercepts for first line

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

Step3: Find intercepts for second line

x-intercept (set $y=0$):
$0 = -\frac{7}{2}x \implies x=0$
Point: $(0, 0)$
Second point (set $x=2$):
$y = -\frac{7}{2}(2) = -7$
Point: $(2, -7)$

Step4: Solve for intersection algebraically

Subtract the two original equations:
$(x + 2y) - (7x + 2y) = 12 - 0$
$-6x = 12 \implies x = -2$
Substitute $x=-2$ into $7x + 2y = 0$:
$7(-2) + 2y = 0 \implies -14 + 2y = 0 \implies y=7$

Answer:

The solution to the system is $x=-2$, $y=7$, or the point $(-2, 7)$.
To graph:

1. For $y = -\frac{1}{2}x + 6$, plot $(12, 0)$ and $(0, 6)$, then draw the line through them.
2. For $y = -\frac{7}{2}x$, plot $(0, 0)$ and $(2, -7)$, then draw the line through them.
3. The lines intersect at $(-2, 7)$.