Understanding the Equation of a Line with Given Intercepts: A Comprehensive Guide

When working with linear equations, it's important to understand how to derive the equation based on given intercepts on the coordinate axes. This article will explore the process of finding the equation of a line that cuts off intercepts of -5 and 3 from the x and y axes, respectively. We'll discuss key concepts, alternative forms, and step-by-step derivations in a way that aligns with Google SEO standards.

Intercept Form of the Equation of a Line

Given the x-intercept as -5 and the y-intercept as 3, we can derive the equation of the line using the intercept form. The intercept form of a line is given by:

[frac{x}{a} frac{y}{b} 1]

Substituting the given intercepts, we get:

[frac{x}{-5} frac{y}{3} 1]

Multiplying through by -15 to clear the denominators, we obtain:

[3x - 5y -15]

This equation represents a line that:

Crosses the x-axis at x -5 Crosses the y-axis at y 3

Slope-Intercept Form of the Equation of a Line

The slope-intercept form of a line is given by:

[y mx b]

where m is the slope and b is the y-intercept. Using the given intercepts, we can find the slope:

[m frac{3 - 0}{0 - (-5)} frac{3}{5}]

Using the point (0, 3) as our starting point and the slope, we can write the equation in slope-intercept form:

[y - 3 frac{3}{5}(x - 0)]

Simplifying, we get:

[y frac{3}{5}x 3]

Alternative Forms of the Equation

Alternatives to the intercept form and slope-intercept form include the standard form of a line, which is given by:

[Ax By C]

where A, B, and C are integers, A cannot be negative, and the greatest common factor (GCF) of A, B, and C must be 1. To convert to the standard form, we can start from the slope-intercept form:

[y frac{3}{5}x 3]

Multiplying through by 5 to clear the fraction, we get:

[5y 3x 15]

Rearranging terms, we obtain:

[3x - 5y -15]

Multiplying through by -1 to get all positive coefficients:

[5x - 3y 15]

This is the standard form of the equation.

Additional Resources

To further explore these concepts and improve your understanding of linear equations, consider checking out the following resources:

Lecture Notes on Linear Equations: Detailed explanations and proofs of various forms of linear equations. Interactive Graphs: Use interactive graphs to visualize the x-intercept and y-intercept and the corresponding line equations. Math Tutoring Websites: Websites that provide step-by-step solutions and practice problems to help you master this topic.

In conclusion, the equation of a line that cuts off intercepts of -5 and 3 from the x and y axes can be represented in multiple forms: intercept form, slope-intercept form, and standard form. Understanding these forms and how to derive them from given intercepts is crucial for solving a wide range of problems in mathematics and related fields.