Understanding the Equation of a Line with Given Slope and Point: A Comprehensive Guide

In this article, we will explore how to determine the equation of a line when given a slope and a point. We will discuss different forms of the equation, such as the slope-intercept form and the standard form, and delve into the mathematical steps involved in finding the equation.

Introduction to the Slope-Point Equation

The equation of a line is typically defined by its slope (m) and its y-intercept (b). The general form of the equation of a line is y mx b. However, when you have the slope and a point that the line passes through, you can use this information to find the specific equation of the line.

Determining the Equation Using the Slope-Intercept Form

Let's start with a specific example where the slope (m) is given as 2/3. To find the equation of the line, we need to determine the y-intercept (b).

Given:

The general form of the equation is:

y mx b

Substitute the given slope and the point (4, 5) into the equation:

5 2/3(4) b

5 8/3 b

b 5 - 8/3 (15/3 - 8/3) 7/3

Therefore, the equation of the line is:

y 2/3x 7/3

Note: If the y-intercept is zero, you can simplify the equation as follows:

y 2/3x

Using the One-Point Form

Another common form to use is the one-point form, given by:

y - y1 m(x - x1)

Substitute the slope (m) and a point (x1, y1) into the equation:

y - 3 2/3(x - 2)

3(y - 3) 2(x - 2)

3y - 9 2x - 4

3y 2x - 4 9

3y 2x 5

Line Example with Given Slope and Point

Suppose we have a line with a slope of 2/3. Let's say the line passes through the point (2, 3). To find the equation, we follow these steps:

3 2/3(2) b

3 4/3 b

b 3 - 4/3 (9 - 4)/3 5/3

Therefore, the equation of the line is:

y 2/3x 5/3

Standard Form of the Equation

The standard form of the equation of a line is given by:

Ax By C 0

To convert the slope-intercept form to the standard form, we can follow these steps:

y 2/3x 5/3

3y 2x 5

2x - 3y 5 0

Conclusion

This guide has provided you with the essential methods to determine the equation of a line given its slope and a point. Whether you are using the slope-intercept form, one-point form, or converting to the standard form, the steps are clear and straightforward. Understanding these concepts is crucial for anyone working with linear equations in mathematics or real-world applications.