Deriving the Equation of a Line Given Its Slope and a Point

In this article, we will explore how to derive the equation of a line using its slope and a point it passes through. Specifically, we will work through a problem where the slope ( m -4 ) and the line passes through the point ((5, -2)). We will cover the point-slope form, the slope-intercept form, and the standard form of the line's equation.

Point-Slope Form

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

[ y - y_1 m(x - x_1) ]

Where ( m ) is the slope and ((x_1, y_1)) is a point on the line. Given ( m -4 ) and the point ((5, -2)), we substitute these values into the formula:

[ y - (-2) -4(x - 5) ]

Simplifying, we get:

[ y 2 -4x 20 ]

Subtracting 2 from both sides, we obtain the equation:

[ y -4x 18 ]

Slope-Intercept Form

The slope-intercept form of a line's equation is:

[ y mx b ]

Here, ( m ) is the slope and ( b ) is the y-intercept. We already have the slope ( m -4 ). To find ( b ), we use the point ((5, -2)) and substitute it into the equation:

[ -2 -4(5) b ]

Simplifying this, we get:

[ -2 -20 b ]

Adding 20 to both sides, we find:

[ b 18 ]

Therefore, the equation is:

[ y -4x 18 ]

Standard Form

The standard form of a line's equation is:

[ ax by c ]

Starting from the slope-intercept form ( y -4x 18 ), we can rearrange it to standard form:

[ 4x y 18 ]

Conclusion

This problem demonstrates the fundamental concepts of linear equations. Understanding the point-slope form, slope-intercept form, and standard form is crucial for solving a variety of linear equation problems. By following these steps, you can derive the equation of a line given its slope and a point it passes through.

Keywords: slope, point-slope form, slope-intercept form, linear equations, algebra