Question

solve the following system of equations graphically on the set of axes below.
$y = -dfrac{1}{6}x + 7$
$y = 2x - 6$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Find intercepts for $y=-\frac{1}{6}x+7$

• y-intercept: Set $x=0$, $y=7$ → point $(0, 7)$
• x-intercept: Set $y=0$, $0=-\frac{1}{6}x+7 \implies \frac{1}{6}x=7 \implies x=42$ (use $x=6$ for a closer point: $y=-\frac{1}{6}(6)+7=6$ → point $(6, 6)$)

Step2: Find intercepts for $y=2x-6$

• y-intercept: Set $x=0$, $y=-6$ → point $(0, -6)$
• x-intercept: Set $y=0$, $0=2x-6 \implies 2x=6 \implies x=3$ → point $(3, 0)$

Step3: Solve algebraically for intersection

Set equations equal:
$$-\frac{1}{6}x+7=2x-6$$
Multiply all terms by 6 to eliminate fraction:
$$-x+42=12x-36$$
Rearrange terms:
$$42+36=12x+x$$
$$78=13x$$
$$x=6$$
Substitute $x=6$ into $y=2x-6$:
$$y=2(6)-6=6$$

Answer:

The solution (intersection point) is $(6, 6)$. To graph:

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