How to Write the Equation of a Line in Point-Slope Form

Writing the equation of a line in point-slope form is a fundamental skill in algebra and is often used in various mathematical and real-world applications. The point-slope form of a line's equation is particularly useful when you know the slope and a point through which the line passes. This article will guide you through the process of writing the equation of a line in point-slope form when given a point and a slope. We will also discuss how to convert this form into slope-intercept and standard forms.

The Point-Slope Form Equation

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

y - y_1  m(x - x_1)

where:

x1 and y1 are the coordinates of a point on the line.

m is the slope of the line.

Problem Solving Through Point-Slope Form

Let's walk through the steps of writing the equation of a line that passes through the point (1, 3) with a slope of -2. This problem can be tackled by simply substituting the known values into the point-slope form equation.

Step 1: Identify the Values

The point (1, 3) means that x1 1 and y1 3. The slope, m -2.

Step 2: Substitute into the Point-Slope Form

Substitute these values into the point-slope form equation:

y - 3  -2(x - 1)

This is the equation of the line in point-slope form. This form is already simplified and does not need further manipulation unless you need to convert it to another form.

Converting to Slope-Intercept Form (y mx b)

The slope-intercept form is y mx b, where b is the y-intercept. To convert the point-slope form to slope-intercept form, you need to distribute and isolate y on one side of the equation.

Starting from:

y - 3  -2(x - 1)

Distribute the -2:

y - 3  -2x   2

Add 3 to both sides:

y  -2x   5

This is the slope-intercept form of the equation, where the y-intercept b 5.

Converting to Standard Form (ax by c)

The standard form of a line's equation is ax by c. To convert the point-slope form to standard form, you can simply rearrange and combine like terms.

Starting from:

y - 3  -2(x - 1)

Distribute the -2:

y - 3  -2x   2

Add 2x to both sides:

2x   y - 3  2

Add 3 to both sides:

2x   y  5

Subtract 5 from both sides to get the standard form:

2x   y - 5  0

This is the standard form of the equation, where a 2, b 1, and c 5.

Understanding the Importance of Point-Slope Form

Point-slope form is particularly useful because it is straightforward to work with when you have a slope and a point. It is a direct representation of the relationship between the slope and a point, making it easier to visualize and understand the line's behavior.

The point-slope form, slope-intercept form, and standard form are all equivalent representations of the same line, each with its own advantages. The choice of which form to use depends on the specific problem or the information you are provided with.

Key Takeaways:

The point-slope form of a line's equation is y - y_1 m(x - x_1) where m is the slope and (x_1, y_1) is a point on the line.

To convert to slope-intercept form, distribute the slope and isolate y.

To convert to standard form, rearrange and combine like terms.