Introduction to Line Equations

Understanding the equation of a line passing through two specific points is a fundamental concept in both mathematics and geometry. This article provides a detailed explanation of how to derive such equations using various forms. We will delve into the point-slope form, slope-intercept form, and standard form, all using the example of a line passing through the points (1, 6) and (3, 2).

Deriving the Equation of the Line

To find the equation of a line passing through the points (1, 6) and (3, 2), we can use several methods. Let's explore how to do this step-by-step:

Method 1: Point-Slope Form

The point-slope form of the equation of a line is given by:

[y - y_1 m(x - x_1)]

where m is the slope of the line, and x_1, y_1 is a point on the line.

Step 1: Calculate the Slope

The slope m can be calculated using the formula:

[m frac{y_2 - y_1}{x_2 - x_1}]

Substituting the given points (1, 6) and (3, 2):

[m frac{2 - 6}{3 - 1} frac{-4}{2} -2]

Step 2: Use the Point-Slope Form

Using the point (1, 6) and the slope m -2 in the point-slope form:

[y - 6 -2(x - 1)]

Simplifying this:

[y - 6 -2x 2]

[y -2x 8]

Method 2: Slope-Intercept Form

Another way to express the equation of a line is in the slope-intercept form:

[y mx c]

where m is the slope and c is the y-intercept.

Step 1: Calculate the Slope

As calculated before:

[m -2]

Step 2: Calculate the Y-Intercept c

We know that the line passes through the point (1, 6). Substituting x 1 and y 6 into the slope-intercept form:

[6 -2(1) c]

[6 -2 c]

[c 8]

Therefore, the equation in slope-intercept form is:

[y -2x 8]

Method 3: Standard Form

The standard form of the equation of a line is:

[Ax By C]

where A, B, and C are constants.

Step 1: Convert from Slope-Intercept Form

Starting from the slope-intercept form:

[y -2x 8]

Rearrange the equation:

[2x y 8]

This is the standard form of the equation.

Conclusion

The equation of the line passing through the points (1, 6) and (3, 2) is:

[y -2x 8] or [2x y 8]

This comprehensive guide has provided you with different methods to find the equation of a line through two given points, utilizing both the point-slope form and slope-intercept form. Understanding these methods can greatly enhance your problem-solving skills in mathematics and geometry.