Question

determine the equation, in slope - intercept form, for the line shown in the graph below:

Explanation:

Step1: Recall slope - intercept form

The slope - intercept form of a line 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 $x = 0$, $y=0$. Substituting into $y=mx + b$, we get $0=m(0)+b$, so $b = 0$.

Step3: Calculate the slope ($m$)

We can use two points on the line. Let's take the points $(0,0)$ and $(1,2)$ (we can also use other points like $(2,4)$ etc.). The formula for slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Using $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(1,2)$, we have $m=\frac{2 - 0}{1 - 0}=\frac{2}{1}=2$. We can verify with another pair of points, say $(2,4)$ and $(3,6)$. $m=\frac{6 - 4}{3 - 2}=\frac{2}{1}=2$.

Step4: Write the equation

Since $m = 2$ and $b = 0$, substituting into $y=mx + b$, we get $y=2x+0$, or simply $y = 2x$.

Answer:

$y = 2x$