Question

graph the line.
y + 3 = \frac{2}{3}(x + 1)

Explanation:

Step1: Rewrite to slope-intercept form

First, isolate $y$ to get $y=mx+b$:

$$\begin{align*} y+3&=\frac{2}{3}(x+1)\\ y&=\frac{2}{3}x + \frac{2}{3} - 3\\ y&=\frac{2}{3}x - \frac{7}{3} \end{align*}$$

Step2: Identify key points

1. Y-intercept: When $x=0$, $y=-\frac{7}{3}\approx-2.33$. This is the point $(0, -\frac{7}{3})$.
2. Point from original form: The equation is in point-slope form $y-y_1=m(x-x_1)$, so we have a point $(-1, -3)$.
3. Second point using slope: Slope $m=\frac{2}{3}$, meaning for every 3 units right ($+3$ in $x$), $y$ increases by 2 ($+2$). From $(-1, -3)$, add $(3,2)$ to get $(2, -1)$.

Step3: Plot and connect points

Plot the points $(-1, -3)$, $(0, -\frac{7}{3})$, and $(2, -1)$, then draw a straight line through them.

Answer:

The line passes through points such as $(-1, -3)$, $(0, -\frac{7}{3})$, and $(2, -1)$, forming a straight line with a slope of $\frac{2}{3}$ and y-intercept at $-\frac{7}{3}$. When graphed, this line rises from left to right, crossing the y-axis near $(0, -2.33)$.