Question

fluency: write the equation of the line in slope - intercept form.

Explanation:

First Graph (Top)

Step1: Identify 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 crosses the y - axis at $(0,1)$, so $b = 1$.

Step3: Calculate the slope ($m$)

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

Step4: Write the equation

Substitute $m = 1$ and $b = 1$ into $y=mx + b$. So the equation is $y=x + 1$.

Second Graph (Bottom)

Step1: Identify 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 crosses the y - axis at $(0,-3)$, so $b=-3$.

Step3: Calculate the slope ($m$)

We can use two points on the line. Let's take $(0,-3)$ and $(-1,0)$ (we can also use other points like $(1,-6)$ and $(0,-3)$). The slope formula is $m = \frac{y_2-y_1}{x_2 - x_1}$. Using $(x_1,y_1)=(0,-3)$ and $(x_2,y_2)=(-1,0)$, we get $m=\frac{0-(-3)}{-1 - 0}=\frac{3}{-1}=- 3$.

Step4: Write the equation

Substitute $m=-3$ and $b = - 3$ into $y=mx + b$. So the equation is $y=-3x-3$.

Answer:

For the top graph: $y = x + 1$
For the bottom graph: $y=-3x - 3$