How to Rewrite an Equation in Slope-Intercept Form: A Comprehensive Guide

Understanding how to rewrite equations in slope-intercept form is crucial for both solving and graphing linear equations. This guide will walk you through the process with practical steps and examples.

Understanding the Slope-Intercept Form

Slope-intercept form is a linear equation that takes the form:

y mx b

Where:

m represents the slope of the line. b represents the y-intercept of the line, the point where the line crosses the y-axis.

From Standard Form to Slope-Intercept Form

Given the equation in standard form:

ax by c

The process involves solving for y. Here’s a step-by-step guide:

Isolate y on one side of the equation: Divide every term by the coefficient of y to get the equation in the form y mx b.

Example:

ax by c

by -ax c

y -a/bx c/b

Using Two Points to Find Slope-Intercept Form

When given two points, you can find the slope and y-intercept to rewrite the equation in slope-intercept form. Follow these steps:

Identify the coordinates of the two points, (x1, y1) and (x2, y2). Calculate the slope m: m (y2 - y1) / (x2 - x1) Use the slope and one of the points to find the y-intercept b: b y1 - mx1 Write the equation in slope-intercept form: y mx b

Example: Two points: (2, 3) and (4, 7)

Identify the coordinates: (x1, y1) (2, 3) and (x2, y2) (4, 7). Calculate the slope: m (7 - 3) / (4 - 2) 4 / 2 2 Use the slope and one of the points to find the y-intercept: b 3 - 2*2 3 - 4 -1 Write the equation: y 2x - 1

Understanding the Slope and Y-Intercept

The slope of a line, m, represents the change in y over the change in x. For example, a slope of 1 means that for a 2-unit increase in x, y increases by 2 units.

The y-intercept, b, is the point where the line crosses the y-axis. It is the value of y when x is 0.

The general form of a linear equation is:

y mx c

Where:

m is the slope of the line. c is the y-intercept, or the point where the line crosses the y-axis.

Conclusion

Mastering the technique of rewriting equations in slope-intercept form is vital for solving linear equations and graphing them. Understanding the slope and y-intercept helps in visualizing the line and predicting its behavior across different points. By following the steps outlined in this guide, you can confidently rewrite any linear equation in the desired form.