How to Find the Equation of a Line in Slope-Intercept Form: A Comprehensive Guide

Understanding how to find the equation of a line using the slope-intercept form is a fundamental skill in geometry and algebra. This article explores the process step-by-step, providing detailed examples to ensure a thorough understanding. Whether you are a student or a professional looking to improve your mathematical skills, this guide will serve as a valuable resource.

What is the Slope-Intercept Form?

The slope-intercept form of a line is expressed as:

y mx b

Where m is the slope of the line and b is the y-intercept, the point at which the line crosses the y-axis.

Steps to Find the Equation of a Line Given Two Points

Step 1: Calculate the Slope

To find the slope m of the line passing through the points (3, -1) and (-1, 5), we use the formula:

m frac{y_2 - y_1}{x_2 - x_1}

Here, x_1 3, y_1 -1, x_2 -1, y_2 5.

Substituting these values into the formula:

m frac{5 - (-1)}{-1 - 3} frac{6}{-4} -frac{3}{2}

Step 2: Use the Point-Slope Form to Find the y-Intercept b

Using the point (3, -1) and the slope m -frac{3}{2}, we can write the equation in the point-slope form:

y - y_1 m(x - x_1)

Substituting the values:

y - (-1) -frac{3}{2}(x - 3)

Simplifying this:

y 1 -frac{3}{2}x frac{9}{2}

Subtract 1 from both sides to get the equation in slope-intercept form:

y -frac{3}{2}x frac{9}{2} - 1

Further simplification:

y -frac{3}{2}x frac{9}{2} - frac{2}{2}

y -frac{3}{2}x frac{7}{2}

Using Vectors to Find the Equation of a Line

The vector vec{AB} (2, 6) helps us understand the direction of the line. This vector is the hypotenuse of a right triangle with base 1 and height 3. If we move one unit to the left from point A, we reach the y-axis at y 0. Moving three units down from there, we reach the y-intercept at y -4. Therefore, the equation of the line in slope-intercept form is:

y 3x - 4

Alternative Solutions

Another way to find the equation of the line through points (1, -1) and (3, 5) is to first calculate the slope:

slope frac{5 - (-1)}{3 - 1} frac{6}{2} 3

Using the point (1, -1) and the slope 3 in the slope-intercept form equation y 3x b, we can find the y-intercept:

-1 3(1) b

Solving for b,

b -1 - 3 -4

Therefore, the equation of the line is:

y 3x - 4

A third method involves calculating the slope as follows:

slope frac{5 - (-1)}{-1 - 3} frac{6}{-4} -frac{3}{2}

This method leads us to the same equation as the first method: