Finding the Equation of a Line Through Two Points: A Step-by-Step Guide

The task of finding the equation of a line that passes through two given points is a fundamental concept in algebra and geometry. In this guide, we will walk through the process using a concrete example, demonstrating how to derive the equation of a line using the slope-intercept form. Let's start with the given points: (2, 5) and (4, 1).

Understanding the Concepts

A non-vertical line can be represented by the equation y ax b. Here, a is the slope (m), and b is the y-intercept (where the line crosses the y-axis).

Step 1: Calculate the Slope

The slope, denoted as m, is the change in y divided by the change in x. It measures the steepness of the line.

Given points: A(2, 5) and B(4, 1)

M change of "y" / change of "x"

M (y2 - y1) / (x2 - x1)

M (1 - 5) / (4 - 2)

M -4 / 2

M -2

Step 2: Substitute the Slope and a Point into the Equation

Now that we have the slope, we can use it along with one of the given points to find the y-intercept, b. We will use the point-slope form of the equation, which is y - y1 m(x - x1). We will use point A(2, 5).

y - y1 m(x - x1)

y - 5 -2(x - 2)

Step 3: Simplify the Equation

Expand and simplify the equation to obtain the slope-intercept form, y mx b.

y - 5 -2(x - 2)

y - 5 -2x 4

y -2x 4 5

y -2x 9

Verification

To ensure our equation is correct, we can verify by plugging the x-values of the given points into the equation and checking if the y-values match.

For x 2:

y -2(2) 9

y -4 9

y 5

This matches the y-value of the first point (2, 5).

For x 4:

y -2(4) 9

y -8 9

y 1

This matches the y-value of the second point (4, 1).

Conclusion

The equation of the line passing through the points (2, 5) and (4, 1) is y -2x 9. This method can be applied to any pair of points to find the equation of the line that passes through them.

Additional Resources

For more information on linear equations and related topics, refer to the following resources:

Khan Academy - Slope-Intercept Form Math is Fun - Equation of a Line Calculatorsoup - Line Equation Calculator