Question

solve the following system of equations graphically on the set of axes below.
$y = -dfrac{1}{6}x + 7$
$y = 2x - 6$

Explanation:

Step1: Find intercepts for first line

For $y = -\frac{1}{6}x + 7$:

• x-intercept: set $y=0$:

$0 = -\frac{1}{6}x + 7 \implies \frac{1}{6}x = 7 \implies x=42$

• y-intercept: set $x=0$: $y=7$

Points: $(42,0)$ and $(0,7)$

Step2: Find intercepts for second line

For $y = 2x - 6$:

• x-intercept: set $y=0$:

$0 = 2x - 6 \implies 2x=6 \implies x=3$

• y-intercept: set $x=0$: $y=-6$

Points: $(3,0)$ and $(0,-6)$

Step3: Find intersection algebraically

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

Answer:

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

1. Plot $(42,0)$ and $(0,7)$, draw a line through them for $y = -\frac{1}{6}x + 7$
2. Plot $(3,0)$ and $(0,-6)$, draw a line through them for $y = 2x - 6$
3. The lines cross at $(6,6)$