Question

graph these equations:
$2x - 3y = 12$
$y = \frac{2}{3}x - 3$

Explanation:

Step1: Rewrite first equation to slope-intercept form

Start with $2x - 3y = 12$. Isolate $y$:
$-3y = -2x + 12$
$y = \frac{2}{3}x - 4$

Step2: Identify key values for first line

Slope $m_1 = \frac{2}{3}$, y-intercept $(0, -4)$.
To find x-intercept, set $y=0$:
$0 = \frac{2}{3}x - 4$
$\frac{2}{3}x = 4$
$x = 6$, so x-intercept $(6, 0)$.

Step3: Identify key values for second line

For $y = \frac{2}{3}x - 3$, slope $m_2 = \frac{2}{3}$, y-intercept $(0, -3)$.
To find x-intercept, set $y=0$:
$0 = \frac{2}{3}x - 3$
$\frac{2}{3}x = 3$
$x = \frac{9}{2} = 4.5$, so x-intercept $(4.5, 0)$.

Step4: Graph the lines

1. For $y = \frac{2}{3}x - 4$: Plot $(0, -4)$ and $(6, 0)$, draw a straight line through them.
2. For $y = \frac{2}{3}x - 3$: Plot $(0, -3)$ and $(4.5, 0)$, draw a straight line through them.

(Note: The lines are parallel, as they have identical slopes.)

Answer:

• Line 1 ($2x-3y=12$ rewritten as $y=\frac{2}{3}x-4$): passes through $(0, -4)$ and $(6, 0)$, slope = $\frac{2}{3}$
• Line 2 ($y=\frac{2}{3}x-3$): passes through $(0, -3)$ and $(4.5, 0)$, slope = $\frac{2}{3}$

The two lines are parallel (no intersection) due to equal slopes.