Introduction

The point-slope form is a fundamental concept in analytic geometry, often used to express the equation of a line when given a point and the slope. This form is especially useful in situations where the traditional slope-intercept form, y mx b, may not be as straightforward or applicable.

The purpose of this article is to explore the point-slope form, its derivation, and practical applications. We will also compare it with the slope-intercept form to clarify its utility in various scenarios.

What is the Point-Slope Form?

The point-slope form of a line is written as:

y - y? m(x - x?)

Here, (x?, y?) is a known point on the line, and m is the slope of the line.

Derivation of the Point-Slope Form

Consider two points (x?, y?) and (x?, y?). The slope m of the line passing through these points is given by:

m (y? - y?) / (x? - x?)

If we know one of the points and the slope, the point-slope form can be used to find the equation of the line.

Let's take the example of the points (2, 1) and (3, 5).

Using the point-slope form:

y - 1 m(x - 2)

We need to find the slope m first:

m (5 - 1) / (3 - 2) 4/1 4

Substituting m back into the point-slope form:

y - 1 4(x - 2)

Simplifying this gives:

y - 1  4x - 8y  4x - 7

Thus, the equation of the line in point-slope form is y - 1 4(x - 2), and in slope-intercept form, it is y 4x - 7.

How to Solve Using Point-Slope Form

Step 1: Identify the Known Point and Slope

Write down the slope m and the coordinates of the reference point (x?, y?).

Step 2: Substitute into the Point-Slope Formula

Replace the slope and the coordinates in the formula:

y - y? m(x - x?)

Step 3: Simplify to Standard Form

Use algebraic manipulation to isolate y and express the equation in standard form y mx b.

Comparison with Other Forms

The point-slope form and the slope-intercept form, y mx b, are closely related. The point-slope form can be derived from the slope-intercept form when a point (x?, y?) on the line is known.

If you know a point (x?, y?) and the slope m, you can use the point-slope form to find the equation of the line:

y - y? m(x - x?)

For instance, if you know a point (2, 1) and the slope is 4, the point-slope form gives:

y - 1 4(x - 2)

Simplifying, you get:

y  4x - 7

Which is the same as the slope-intercept form.

Point slope is advantageous when the y-intercept is not readily available, as it requires only a point and the slope to define the line.

Conclusion

The point-slope form is a versatile and powerful tool in analytic geometry. It is particularly useful when the slope is known but the y-intercept is not, or when working with a specific point on the line. While it may seem like an additional concept to learn, it offers a convenient way to express and manipulate line equations.