Understanding the Equation of a Line Given Slope and Y-Intercept

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

Equation: y mx b

Where:

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

This guide will help you derive the equation of a line with a given slope and y-intercept, and provide examples to illustrate the process.

Deriving the Slope-Intercept Form

To find the equation of a line with slope m 2/5 and y-intercept b -3, follow these steps:

Start with the equation of a line in slope-intercept form: y mx b. Substitute the given slope m 2/5 into the equation: y (2/5)x b. Substitute the given y-intercept b -3 into the equation: y (2/5)x - 3.

The equation of the line is therefore:

Final Equation: y (2/5)x - 3

Alternative Methods to Find the Equation

Substituting a Point on the Line

Suppose you know the line passes through the point (2, -3). You can use this information to find the equation:

Write the equation in the form y - y_1 m(x - x_1), where (x_1, y_1) is a point on the line and m is the slope. Substitute the slope m 2/5 and the point (2, -3): y - (-3) (2/5)(x - 2)? Expand and simplify: y 3 (2/5)(x - 2). Clear the fraction by multiplying everything by 5: 5(y 3) 2(x - 2). Simplify: 5y 15 2x - 4. Rearrange to the standard form: 2x - 5y 19. Convert to the slope-intercept form: y (2/5)x - 19/5.

The equation is now in the desired slope-intercept form: y (2/5)x - 19/5.

Using the Slope and Intercept Form

Another method involves using the form x/a y/b 1, where a and b are the x-intercept and y-intercept, respectively.

Start with the equation x - 2 (2/5)(y 3). Multiply through to clear the fraction: 5(x - 2) 2(y 3). Expand and simplify: 5x - 10 2y 6. Reorganize to standard form: 5x - 2y - 16 0. Divide by 16 to convert to slope-intercept form: x/(16/5) y/(-8) 1.

This provides the equation in slope-intercept form: x/(16/5) y/(-8) 1.

Conclusion

By following these steps and methods, you can derive the equation of a line given its slope and intercept. Understanding these concepts is fundamental in algebra and calculus, and will be useful in more advanced topics such as graphing and linear programming.

Keywords: slope-intercept form, equation of a line, y-intercept