Q.Write the equation of the line through the points $(1, -1)$ and $(3, 5)$.

Concept: Point-slope form from two points

We need the slope and then use either point to write the equation.

Step 1: Find the slope using $m = \frac{y_2 - y_1}{x_2 - x_1}$.

$m = \frac{5 - (-1)}{3 - 1} = \frac{6}{2} = 3$ …

Understanding the Problem

A line is completely determined by any two points it passes through. Our task is to translate the geometric fact that the line contains $(1, -1)$ and $(3, 5)$ into an algebraic equation of the form $y = mx + c$ or one of its equivalent forms.

The strategy: calculate how steeply the line rises (its slope), then anchor that slope at one of the known points.

Solution

1. Calculate the slope

The slope $m$ measures the rate of change of $y$ with respect to $x$. Between any two points $(x_1, y_1)$ and $(x_2, y_2)$, it is given by:

Using $(1, -1)$ as $(x_1, y_1)$ and $(3, 5)$ as $(x_2, y_2)$:

The line rises 3 units vertically for every 1 unit horizontally.

2. Use point-slope form

Now that we know the slope is $m = 3$, we can write the equation using either point. The point-slope form is:

Using the point $(1, -1)$:

3. Verify with the second point

A quick check: does $(3, 5)$ satisfy $y = 3x - 4$? …