Question

solve the following system of equations graphically on the set of axes below.
$y = \frac{2}{3}x - 7$
$y = -\frac{1}{3}x - 4$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Find intercepts for $y=\frac{2}{3}x-7$

• y-intercept: Set $x=0$, so $y=\frac{2}{3}(0)-7=-7$. Point: $(0, -7)$
• x-intercept: Set $y=0$, so $0=\frac{2}{3}x-7 \implies \frac{2}{3}x=7 \implies x=\frac{21}{2}=10.5$. Point: $(10.5, 0)$

Step2: Find intercepts for $y=-\frac{1}{3}x-4$

• y-intercept: Set $x=0$, so $y=-\frac{1}{3}(0)-4=-4$. Point: $(0, -4)$
• x-intercept: Set $y=0$, so $0=-\frac{1}{3}x-4 \implies \frac{1}{3}x=-4 \implies x=-12$. Point: $(-12, 0)$

Step3: Solve algebraically for intersection

Set equations equal:
$$\frac{2}{3}x-7 = -\frac{1}{3}x-4$$
Add $\frac{1}{3}x$ to both sides:
$$x - 7 = -4$$
Add 7 to both sides:
$$x=3$$
Substitute $x=3$ into $y=\frac{2}{3}x-7$:
$$y=\frac{2}{3}(3)-7=2-7=-5$$

Answer:

The solution (intersection point of the two lines) is $(3, -5)$. To graph:

1. Plot the line $y=\frac{2}{3}x-7$ using points $(0, -7)$ and $(10.5, 0)$, then draw a line through them.
2. Plot the line $y=-\frac{1}{3}x-4$ using points $(0, -4)$ and $(-12, 0)$, then draw a line through them.
3. The lines cross at $(3, -5)$.