Question

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

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 y - intercept is the point where the line crosses the y - axis. From the graph, the line crosses the y - axis at $(0,1)$, so $b = 1$.

Step3: Calculate the slope ($m$)

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points on the line. For example, $(0,1)$ and $(1,2)$.
Using the slope formula: $m=\frac{2 - 1}{1 - 0}=\frac{1}{1}=1$. We can also check with other points, like $(2,3)$ and $(3,4)$. $m=\frac{4 - 3}{3 - 2}=\frac{1}{1}=1$. So the slope $m = 1$.

Step4: Write the equation

Substitute $m = 1$ and $b = 1$ into the slope - intercept form $y=mx + b$. We get $y=1x + 1$, which simplifies to $y=x + 1$.

Answer:

$y = x + 1$