To find the equation of the line passing through the points $(-1, -1)$ and $(3, 7)$, we can use the point-slope form of a line equation, which is:

$y - y_1 = m(x - x_1)$

Where:

• $m$ is the slope of the line
• $(x_1, y_1)$ is a point on the line

Step 1: Find the slope $m$

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of the points $(-1, -1)$ and $(3, 7)$:

$m = \frac{7 - (-1)}{3 - (-1)} = \frac{7 + 1}{3 + 1} = \frac{8}{4} = 2$

Step 2: Use the point-slope form

Now that we have the slope $m = 2$, we can use either point $(-1, -1)$ or $(3, 7)$ in the point-slope form. Let's use the point $(-1, -1)$:

$y - (-1) = 2(x - (-1))$

Simplifying:

$y + 1 = 2(x + 1)$

Step 3: Simplify to slope-intercept form

Now, distribute the $2$ on the right-hand side:

$y + 1 = 2x + 2$

Subtract 1 from both sides to get the equation in slope-intercept form ($y = mx + b$):

$y = 2x + 1$

Final Answer:

The equation of the line passing through the points $(-1, -1)$ and $(3, 7)$ is:

$y = 2x + 1$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you find the equation of a line in point-slope form?
2. What is the meaning of the slope $m$ in a line equation?
3. How do you convert a line equation from point-slope form to slope-intercept form?
4. What is the process for finding the equation of a line if given a slope and a point?
5. Can you derive the equation of a line if given its slope and the y-intercept?

Tip: Remember that the slope of a line describes how steep the line is, and it's calculated as the change in $y$ over the change in $x$.