Equation of a Line with Gradient -1/3 and Y-Intercept -4: A Comprehensive Guide

Understanding the equation of a line is essential for students and professionals in mathematics and related fields. This guide will explain how to find the equation of a line given its gradient and y-intercept, specifically when the gradient is -1/3 and the y-intercept is -4.

Introduction to the Equation of a Line

The general form of the equation of a line is y ax b, where:

a is the slope (gradient) of the line.

b is the y-intercept of the line.

Given Data

For this example, we are given:

Gradient (slope) a -1/3

Y-intercept b -4

Formulating the Equation

To find the equation of the line, we will use the information given:

Identify the slope (gradient) a -1/3

Identify the y-intercept b -4

Substitute these values into the general form y ax b

Substituting the values, we get:

y -1/3x - 4

Alternative Forms of the Equation

The equation can also be written in alternative forms for clarity or specific requirements:

Standard form: y -x/3 - 4

The equation can be rearranged further, for example: y (-x - 12)/3

Additional Insights

Let's explore an alternative approach to formulating the equation using the point-slope form of a line:

Point-slope form: y - y1 m(x - x1)

We know the gradient m -1/3

The point on the line is (x1, y1) (0, -4)

Substituting the point and the slope into the point-slope form, we get:

y 4 -1/3(x - 0)

After simplification:

y -x/3 - 4

This confirms our earlier equation in alternative form.

Equation in Standard Form

The standard form of a linear equation is ax by c. To express our equation in this form:

Start with the equation: y -1/3x - 4

Multiply both sides by 3 to eliminate the fraction:

3y -x - 12

Reorganize to the standard form:

x 3y -12

Conclusion

In summary, the equation of the line with gradient -1/3 and y-intercept -4 is:

y -1/3x - 4

This can be expressed in various forms, including:

y -x/3 - 4

y (-x - 12)/3

x 3y -12

Understanding these different representations is crucial for various applications in algebra, geometry, and beyond.