To find the equation of the line passing through the points $(-2, 7)$ and $(3, -3)$, we use the slope-intercept form of a line, $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Step 1: Find the slope ($m$)

The slope is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the given points $(-2, 7)$ and $(3, -3)$: $m = \frac{-3 - 7}{3 - (-2)} = \frac{-10}{5} = -2$

Step 2: Use the slope-intercept form

The equation of the line is $y = mx + b$. Substituting $m = -2$, the equation becomes: $y = -2x + b$

Step 3: Solve for $b$

Use one of the given points to find $b$. Let’s use $(-2, 7)$: $7 = -2(-2) + b$ $7 = 4 + b$ $b = 3$

Final Equation

The equation of the line is: $y = -2x + 3$

Would you like a detailed breakdown or have questions?

Expanding Questions:

1. What are the steps to find the slope of a line from two points?
2. How do you determine the $y$-intercept once the slope is known?
3. Can this method be extended to vertical or horizontal lines? Why or why not?
4. What happens if the slope is zero or undefined?
5. How do you convert the equation of a line to standard form?

Tip:

When finding the slope, always check the order of subtraction to avoid sign errors.