Question

write the equation of the line in fully simplified slope-intercept form.

Explanation:

Step1: Identify slope-intercept form

Slope - intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step2: Find the y - intercept ($b$)

The line crosses the y - axis at $(0,2)$, so $b = 2$.

Step3: Calculate the slope ($m$)

We can use two points on the line. Let's take $(0,2)$ and $(- 3,0)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
Substitute $x_1 = 0,y_1 = 2,x_2=-3,y_2 = 0$ into the formula:
$m=\frac{0 - 2}{-3 - 0}=\frac{-2}{-3}=\frac{2}{3}$? Wait, no, let's check another pair. Let's take $(3,4)$ and $(0,2)$. Then $m=\frac{4 - 2}{3 - 0}=\frac{2}{3}$? Wait, no, wait the line passes through $(-3,0)$ and $(0,2)$? Wait, no, when $x = 0$, $y = 2$; when $x=3$, $y = 4$? Wait, no, let's count the rise over run. From $(0,2)$ to $(3,4)$, the rise is $4 - 2=2$, the run is $3 - 0 = 3$, so slope $m=\frac{2}{3}$? Wait, no, wait the line also passes through $(-3,0)$. Let's check the slope between $(-3,0)$ and $(0,2)$: $\frac{2-0}{0 - (-3)}=\frac{2}{3}$. Yes, that's correct. Wait, but wait, let's check another point. When $x = 3$, $y=4$: $\frac{4 - 2}{3-0}=\frac{2}{3}$. So slope $m=\frac{2}{3}$? Wait, no, wait the line in the graph: when $x = 0$, $y = 2$; when $x=-3$, $y = 0$; when $x = 3$, $y=4$. So the slope is $\frac{2}{3}$? Wait, no, wait maybe I made a mistake. Wait, let's take two points: $(0,2)$ and $(3,4)$. The change in $y$ is $4 - 2 = 2$, change in $x$ is $3-0 = 3$, so slope $m=\frac{2}{3}$. Then the equation is $y=\frac{2}{3}x + 2$? Wait, no, wait when $x=-3$, $y=\frac{2}{3}\times(-3)+2=-2 + 2=0$, which matches the point $(-3,0)$. So that's correct.

Wait, but let's re - check. The slope - intercept form is $y=mx + b$, $b = 2$, $m=\frac{2}{3}$. So the equation is $y=\frac{2}{3}x+2$? Wait, no, wait maybe I messed up the slope. Let's take $(0,2)$ and $(3,4)$: $\frac{4 - 2}{3-0}=\frac{2}{3}$. Correct. So the equation is $y=\frac{2}{3}x + 2$? Wait, no, wait the line in the graph: when $x = 1$, $y=\frac{2}{3}(1)+2=\frac{2 + 6}{3}=\frac{8}{3}\approx2.666$, but in the graph, at $x = 3$, $y = 4$. Wait, when $x = 3$, $\frac{2}{3}\times3+2=2 + 2=4$, which is correct. At $x=-3$, $\frac{2}{3}\times(-3)+2=-2 + 2=0$, correct. At $x = 0$, $y = 2$, correct. So the equation is $y=\frac{2}{3}x + 2$? Wait, no, wait maybe the slope is $\frac{1}{2}$? Wait, no, let's count the grid. Each grid square is 1 unit. From $(0,2)$ to $(3,4)$: up 2, right 3. So slope is $\frac{2}{3}$. So the equation is $y=\frac{2}{3}x+2$.

Wait, no, wait another way: the y - intercept $b = 2$. Let's find two points: $(0,2)$ and $(3,4)$. The slope $m=\frac{4 - 2}{3-0}=\frac{2}{3}$. So the equation is $y=\frac{2}{3}x + 2$.

Answer:

$y=\frac{2}{3}x + 2$