What is the Equation of a Line Passing Through Two Given Points?

When dealing with linear equations, it's common to need to find the equation of a line that passes through two given points. This article will guide you through the process of finding the equation of a line using the slope-intercept form, a fundamental concept in analytic geometry and algebra.

General Concept

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

y mx b

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

Step-by-Step Guide

Step 1: Calculate the Slope (m)

The slope of a line can be calculated using the following formula:

m (y? - y?) / (x? - x?)

Where:

(x?, y?) and (x?, y?) are the coordinates of the two points.

Example

Given the points (-1, 1) and (3, 5), we first calculate the slope:

m (5 - 1) / (3 - (-1)) 4 / 4 1

Step 2: Use the Point-Slope Form to Find the Equation

Once the slope is known, the point-slope form of the equation can be used. This form is given by:

y - y? m(x - x?)

Using one of the points, let's use (-1, 1):

y - 1 1(x - (-1))

This simplifies to:

y - 1 x 1

Step 3: Rearrange to Slope-Intercept Form

To express the equation in the standard slope-intercept form (y mx b), we need to solve for y:

y - 1 x 1

Add 1 to both sides:

y x 2

Thus, the equation of the line passing through the points (-1, 1) and (3, 5) is:

y x 2

Additional Examples

Example 1: Equation of Line AB

Let's consider the vector AB which is 26 213. The line coincidental with this vector is the hypotenuse of a right triangle whose base and height are in the ratio 1:3. If we use the components of AB, we can see that run and rise are either both positive or both negative. Thus, when we run -1 unit from point A, we reach the y-axis at (0, -1). Going down 3 units from there, we reach the y-intercept at (0, -4). Hence, the equation of line AB in slope-intercept form is:

y 3x - 4

Example 2: Equation of Line Through Points 1, -1 and 3, 5

To find the equation of the line through the points (1, -1) and (3, 5), we first calculate the slope:

m (5 - (-1)) / (3 - 1) 6 / 2 3

Now, using the general form of the equation and substituting one of the points (1, -1), we get:

-1 3(1) b

Solving for b:

b -4

Thus, the equation of the line is:

y 3x - 4

This method can be applied to verify the correctness of the equation by plugging in the given points.

Conclusion

The process of finding the equation of a line through two points can be done using the slope-intercept form. By identifying the slope and the y-intercept, you can express the line in the form y mx b. This technique is widely used in both theoretical and practical applications in mathematics and related fields.