complete the point-slope equation of the line through and .

Guri Bee

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21-10-2024

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Finding the Equation of a Line: Point-Slope Form Explained

In the realm of algebra, understanding the equation of a line is fundamental. One of the most common forms for representing a line is the point-slope form. This article will delve into the point-slope form and demonstrate how to use it to determine the equation of a line given two points.

Understanding the Point-Slope Form

The point-slope form of a linear equation is:

**y - y₁ = m(x - x₁) **

where:

• m represents the slope of the line.
• **(x₁, y₁) ** is a point that lies on the line.

This form is particularly useful because it directly incorporates the slope and a specific point on the line.

Example: Finding the Equation of a Line Given Two Points

Let's say we are given two points: (2, 3) and (5, 9). Our goal is to determine the equation of the line that passes through these points.

1. Calculate the Slope (m):

   The slope of a line is the ratio of the change in y-coordinates to the change in x-coordinates. Using our two points:

   m = (9 - 3) / (5 - 2) = 6 / 3 = 2

2. Choose a Point:

   We can choose either point (2, 3) or (5, 9). Let's select (2, 3) for this example.

3. Plug in the Values:

   Now we plug the slope (m = 2) and the chosen point (x₁ = 2, y₁ = 3) into the point-slope form:

   y - 3 = 2(x - 2)

4. Simplify (Optional):

   While this is a perfectly valid equation, you might prefer to rearrange it into slope-intercept form (y = mx + b) by distributing the 2 and isolating y:

   y - 3 = 2x - 4 y = 2x - 1

Therefore, the equation of the line passing through (2, 3) and (5, 9) is y - 3 = 2(x - 2) or y = 2x - 1.

Key Takeaways

• The point-slope form is a convenient way to represent the equation of a line given its slope and a point on the line.
• To find the equation of a line using this form, you need to calculate the slope and choose one of the points on the line.
• You can easily convert the point-slope form into slope-intercept form by simplifying the equation.

This article provides a basic understanding of using the point-slope form to find the equation of a line. Remember, practice is key to mastering this concept!