Finding the Equation of a Line with a Given Point and Slope

This article discusses how to find the equation of a line that passes through a specific point and has a given slope. The process involves the use of both the point-slope form and the slope-intercept form of a line's equation. We'll walk through the steps using a specific example.

The Point-Slope Form and Its Application

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

y - y1 m(x - x1)

where:

m is the slope (gradient) of the line. (x1, y1) is a point the line passes through.

Let's apply this to the specific example given: a line that passes through (13, 4) with a slope of -2.

Example 1

Identify the values: m -2, x1 13, y1 4. Substitute these values into the point-slope form:

y - 4 -2(x - 13)

Simplify the equation:

y - 4 -2x 26

Further simplify to get the equation in slope-intercept form:

y -2x 30

This is the equation of the line in slope-intercept form, y mx b.

Using the General Equation of a Line

Another way to express the equation of a line is through the general form:

y mx c

Given that the line passes through the point (13, 4) and has a slope of -2, we can substitute these values into the equation to find c (the y-intercept).

Example 2

Substitute the values into the general form:

4 -2(13) c

Solve for c:

c 4 26 30

Therefore, the equation of the line is:

y -2x 30

Conclusion

In summary, we have two methods to find the equation of a line given a point and a slope:

The point-slope form: y - y1 m(x - x1) The slope-intercept form: y mx c

Both methods yield the same result: the equation of the line is y -2x 30 or y -2x - 30 if you simplify the expression differently.

Additional Resources

For further reading on this topic, you can explore the following resources:

Math Is Fun - Point-Slope Form Khan Academy - Slope-Intercept Form