Question

1. write an equation in slope-intercept form to represent the graph shown.

Explanation:

Step1: Recall slope - intercept form

The slope - intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

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

The line passes through the origin $(0,0)$, so when $x = 0$, $y=0$. Thus, $b = 0$.

Step3: Calculate the slope ($m$)

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. We can use two points on the line. Let's use the points $(0,0)$ and $(9,60)$ (we can also use other points like $(3,20)$ etc.).
Using $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(9,60)$, we have $m=\frac{60 - 0}{9 - 0}=\frac{60}{9}=\frac{20}{3}$? Wait, no, wait. Wait, looking at the graph, when $x = 3$, $y = 20$; when $x=6$, $y = 40$; when $x = 9$, $y=60$. Wait, let's recalculate. Let's take $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(3,20)$. Then $m=\frac{20 - 0}{3 - 0}=\frac{20}{3}$? No, wait, $20\div3$ is not correct. Wait, when $x = 1$, $y=\frac{20}{3}$? No, wait, maybe I made a mistake. Wait, looking at the graph, the line goes through $(0,0)$ and $(9,60)$. Wait, $60\div9=\frac{20}{3}\approx6.666$, but wait, when $x = 3$, $y = 20$, $20\div3=\frac{20}{3}$, when $x = 6$, $y = 40$, $40\div6=\frac{20}{3}$, when $x=9$, $y = 60$, $60\div9=\frac{20}{3}$. Wait, but maybe a simpler way: the slope $m=\frac{\Delta y}{\Delta x}$. From $(0,0)$ to $(3,20)$, $\Delta y=20$, $\Delta x = 3$, so $m=\frac{20}{3}$? No, wait, no, wait the grid: each square is, let's see, the y - axis has marks at 10,20,30,... and x - axis at 1,2,3,... Wait, when $x = 3$, $y = 20$; $x = 6$, $y = 40$; $x=9$, $y = 60$. So the rate of change is $\frac{20}{3}$? Wait, no, wait $y$ increases by 20 when $x$ increases by 3, so $m=\frac{20}{3}$. But wait, maybe I misread the graph. Wait, no, wait the line passes through $(0,0)$ and $(9,60)$. So slope $m=\frac{60}{9}=\frac{20}{3}$. But wait, let's check with another pair. $(3,20)$: $\frac{20}{3}\times3=20$, correct. $(6,40)$: $\frac{20}{3}\times6 = 40$, correct. $(9,60)$: $\frac{20}{3}\times9=60$, correct.
Wait, but wait, maybe I made a mistake. Wait, the slope - intercept form is $y=mx + b$. Since $b = 0$ (passes through origin), and $m=\frac{20}{3}$? No, wait, no, wait the graph: when $x = 1$, $y=\frac{20}{3}\approx6.666$, but looking at the graph, the line at $x = 3$ is at $y = 20$, $x = 6$ at $y = 40$, $x=9$ at $y = 60$. So the slope is $\frac{60}{9}=\frac{20}{3}$. So the equation is $y=\frac{20}{3}x$? Wait, no, wait, maybe I miscalculated. Wait, let's take two points: $(0,0)$ and $(3,20)$. The slope $m=\frac{20 - 0}{3 - 0}=\frac{20}{3}$. So the equation is $y=\frac{20}{3}x$? Wait, but let's check with $x = 9$, $y=\frac{20}{3}\times9 = 60$, which matches the graph. Alternatively, maybe the slope is $\frac{20}{3}$, but wait, maybe I made a mistake in the points. Wait, another way: the line passes through $(0,0)$ and $(9,60)$. So the slope $m=\frac{60}{9}=\frac{20}{3}$. So the equation is $y=\frac{20}{3}x$? Wait, no, wait, maybe the slope is $\frac{20}{3}$? Wait, but let's see, when $x = 3$, $y = 20$, so $20=\frac{20}{3}\times3$, which is correct. So the equation in slope - intercept form is $y=\frac{20}{3}x$? Wait, no, wait, maybe I messed up. Wait, the graph: the y - axis is from 0 to 100, x - axis from 0 to 10. The line goes through (0,0), (3,20), (6,40), (9,60). So the slope is $\frac{20}{3}$, and y - intercept is 0. So the equation is $y=\frac{20}{3}x$? Wait, but maybe the slope is $\frac{20}{3}$, so the equation is $y=\frac{20}{3}x$.

Wait, no, wait, maybe I made a mistake in the slope calculation. Wait, let's take $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(3,20)$. Then $m=\frac{20 - 0}{3 - 0}=\frac{20}{3}$. So the equation is $y=\frac{20}{3}x$. But let's check with $x = 1$, $y=\frac{20}{3}\approx6.666$, which is on the line (since the line passes through (0,0) and (3,20), so at $x = 1$, it should be $\frac{20}{3}$). So the equation is $y=\frac{20}{3}x$.

Wait, but maybe the slope is $\frac{20}{3}$, so the equation is $y=\frac{20}{3}x$.

Answer:

$y=\frac{20}{3}x$