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. From the graph, the line crosses the y - axis at $(0,6)$, so $b = 6$.

Step2: Calculate the slope ($m$)

We can use two points on the line. We know $(0,6)$ and another point, say $(-6,1)$ (from the graph: when $x=-6$, $y = 1$). The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(0,6)$ and $(x_2,y_2)=(-6,1)$. Then $m=\frac{1 - 6}{-6-0}=\frac{-5}{-6}=\frac{5}{6}$? Wait, no, wait. Wait, another point: let's take $(0,6)$ and $(6,11)$. Then $m=\frac{11 - 6}{6 - 0}=\frac{5}{6}$? Wait, no, wait, when $x = 6$, $y = 11$? Wait, no, the grid: from $(0,6)$ to $(6,11)$: the change in $y$ is $11 - 6=5$, change in $x$ is $6 - 0 = 6$, so $m=\frac{5}{6}$? Wait, no, wait, let's check with $(-6,1)$ and $(0,6)$: $\frac{6 - 1}{0-(-6)}=\frac{5}{6}$. Yes.

Wait, but wait, let's re - check. The line passes through $(0,6)$ and $(-6,1)$? Wait, when $x=-6$, $y = 1$? Let's count the grid. From $(0,6)$, moving left 6 units (to $x=-6$) and down 5 units (to $y = 1$), so the slope is $\frac{5}{6}$. Then the equation is $y=\frac{5}{6}x + 6$? Wait, no, wait, maybe I made a mistake. Wait, let's take two points: $(0,6)$ and $(6,11)$. The difference in $y$: $11 - 6 = 5$, difference in $x$: $6 - 0=6$, so slope $m=\frac{5}{6}$. So the equation is $y=\frac{5}{6}x+6$? Wait, but let's check with $x=-6$: $y=\frac{5}{6}\times(-6)+6=-5 + 6 = 1$, which matches the point $(-6,1)$. Yes.

Wait, but wait, maybe I misread the points. Let's take $(0,6)$ and $(-6,1)$: slope $m=\frac{1 - 6}{-6-0}=\frac{-5}{-6}=\frac{5}{6}$. Correct. So the slope $m=\frac{5}{6}$ and $b = 6$. So the equation in slope - intercept form is $y=\frac{5}{6}x + 6$? Wait, no, wait, wait, when $x = 6$, $y=\frac{5}{6}\times6+6=5 + 6 = 11$, which matches the point $(6,11)$ (from the graph: at $x = 6$, $y = 11$). Yes.

Wait, but let's check another way. The y - intercept $b = 6$ (since the line crosses the y - axis at $(0,6)$). Then, using the slope formula between $(0,6)$ and $(6,11)$: $m=\frac{11 - 6}{6 - 0}=\frac{5}{6}$. So the equation is $y=\frac{5}{6}x+6$.

Answer:

$y=\frac{5}{6}x + 6$