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 $(0, - 1)$? Wait, no, wait. Wait, let's check the points again. Wait, when $x = 0$, looking at the line, when $x = 0$, $y=-1$? Wait, no, wait, maybe I made a mistake. Wait, let's take two points on the line. Let's take $(0, - 1)$ and $(1,1)$? Wait, no, when $x = 1$, $y = 1$? Wait, no, let's look at the graph again. Wait, the line passes through $(0, - 1)$? Wait, no, when $x=0$, the y - coordinate is - 1? Wait, no, maybe I misread. Wait, let's take two clear points. Let's take $(1,1)$ and $(2,3)$. Wait, the difference in $y$ is $3 - 1=2$, difference in $x$ is $2 - 1 = 1$. So slope $m=\frac{2}{1}=2$? Wait, no, wait, let's check another pair. Let's take $(0, - 1)$ and $(1,1)$. The change in $y$ is $1-(-1)=2$, change in $x$ is $1 - 0=1$, so slope $m = 2$. And the y - intercept $b$: when $x = 0$, $y=-1$? Wait, no, when $x = 0$, the line is at $y=-1$? Wait, but let's check the graph again. Wait, the line passes through $(0, - 1)$? Wait, no, maybe I made a mistake. Wait, let's take the point $(0, - 1)$ and $(1,1)$. So using $y=mx + b$, when $x = 0$, $y=b$, so $b=-1$? Wait, no, if $x = 1$, $y = 1$, then $1=m(1)+b$. If $m = 2$, then $1 = 2(1)+b\Rightarrow b=1 - 2=-1$. So the equation would be $y = 2x-1$? Wait, no, wait, let's check another point. Let's take $x = 2$, then $y=2(2)-1 = 3$, which matches the point $(2,3)$ (from the graph, when $x = 2$, $y = 3$). And $x = 3$, $y=2(3)-1=5$, which also matches. So the slope $m = 2$, y - intercept $b=-1$.

Wait, maybe I misread the y - intercept. Let's look at the graph again. The line crosses the y - axis at $(0, - 1)$? Wait, no, when $x = 0$, the y - coordinate is - 1. So the slope - intercept form is $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step1: Calculate the slope ($m$)

We can use two points on the line. Let's take the points $(0, - 1)$ and $(1,1)$. The formula for slope is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Substituting $y_2 = 1$, $y_1=-1$, $x_2 = 1$, $x_1 = 0$:
$m=\frac{1-(-1)}{1 - 0}=\frac{2}{1}=2$

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

The y - intercept is the value of $y$ when $x = 0$. From the point $(0, - 1)$, when $x = 0$, $y=-1$. So $b=-1$.

Step3: Write the equation in slope - intercept form

The slope - intercept form is $y=mx + b$. Substituting $m = 2$ and $b=-1$, we get $y = 2x-1$.

Answer:

$y = 2x-1$