Finding the Equation of a Line: Slope-Intercept Form Given a Point and Slope

When you have a point and the slope of a line, you can find the equation of that line in slope-intercept form. This article will guide you through the process with clear steps and examples. We cover the slope-intercept form, point-slope form, and methods to find the y-intercept given a point and slope.

Slope-Intercept Form of a Line

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

y mx b

m is the slope of the line. b is the y-intercept, the point where the line crosses the y-axis (where x 0).

To find the slope-intercept form of a line when you know a point and the slope, you can use the formula y mx b. If you have the slope m and a point (x_0, y_0), you can substitute these into the equation to solve for b.

Steps to Find the Equation of a Line

Given a slope m and a point (x_0, y_0), follow these steps:

Write the slope-intercept form: y mx b. Substitute the slope m into the equation: y mx b. Substitute the point (x_0, y_0) into the equation: y_0 mx_0 b. Solve the equation for b: b y_0 - mx_0. Substitute b back into the slope-intercept form: y mx (y_0 - mx_0).

For example, if you have a point (4, 5) and a slope of 0.75, you can find the equation as follows:

y 0.75x b

Substitute the point (4, 5) into the equation:

5 0.75(4) b

5 3 b

b 2

The equation in slope-intercept form is:

y 0.75x 2

Point-Slope Form and Slope-Intercept Form

The point-slope form of a line is:

y - y_1 m(x - x_1)

Start with the point-slope form and solve for y to get the slope-intercept form:

y - y_1 m(x - x_1) y m(x - x_1) y_1 y mx - mx_1 y_1 y mx (y_1 - mx_1)

If the point is the y-intercept (b), the equation simplifies to y mx b.

Conclusion

By understanding and applying the slope-intercept form, point-slope form, and the relationship between the point and the line, you can effectively find the equation of a line given a point and its slope. This method is widely used in geometry, algebra, and various fields of science and engineering to model and analyze linear relationships.