How to Find the Linear Equation?

To find the equation of a line, you primarily need two pieces of information: the slope of the line and a point on the line. There are two common forms used to express a linear equation: slope-intercept form and point-slope form.

Using Slope-Intercept Form (y = mx + b)

This method is useful when you know the slope (m) and the y-intercept (b) of the line. The y-intercept is the point where the line crosses the y-axis (x=0).

1. Identify the slope (m): Determine the rate of change of the line (rise over run).
2. Identify the y-intercept (b): Find the point where the line crosses the y-axis.
3. Substitute m and b into the equation: Plug the values of m and b into the equation y = mx + b.

Example:

If the slope (m) is 2 and the y-intercept (b) is -3, the linear equation is: y = 2x - 3.

Using Point-Slope Form (y - y1 = m(x - x1))

This method is useful when you know the slope (m) and any point (x1, y1) on the line.

1. Identify the slope (m): Determine the rate of change of the line (rise over run). If you have two points, you can calculate the slope using the formula: m = (y2 - y1) / (x2 - x1).
2. Identify a point (x1, y1): Choose any point that lies on the line.
3. Substitute m, x1, and y1 into the equation: Plug the values of m, x1, and y1 into the equation y - y1 = m(x - x1).
4. Simplify (Optional): You can simplify the equation into slope-intercept form (y = mx + b) if desired.

Example:

If the slope (m) is -1 and the point (x1, y1) is (4, 2), the linear equation is:

y - 2 = -1(x - 4)

Simplifying to slope-intercept form:

y - 2 = -x + 4

y = -x + 6

Finding the Slope from Two Points

If you are given two points on the line, (x1, y1) and (x2, y2), you first need to calculate the slope:

m = (y2 - y1) / (x2 - x1)

Then, you can use either point-slope form or calculate the y-intercept using one of the points and the slope, and then use slope-intercept form.

Example:

Given points (1, 5) and (3, 9):

1. Calculate the slope: m = (9 - 5) / (3 - 1) = 4 / 2 = 2
2. Use point-slope form with point (1, 5): y - 5 = 2(x - 1)
3. Simplify to slope-intercept form: y - 5 = 2x - 2 => y = 2x + 3

Summary

Finding the equation of a line involves identifying the slope and a point on the line, then using either slope-intercept form or point-slope form to write the equation. Understanding these two methods provides the flexibility to find linear equations from various types of given information.