Understanding the Line Equation Through Given Points

Finding the equation of a line that passes through specific points is a fundamental concept in mathematics, often used in geometry, physics, and engineering. This article will walk you through the process of determining the equation of a line that passes through the points (2, -1) and (3, 2). This guide will be beneficial for students, educators, and professionals who need to understand linear equations and their applications. You will learn about the slope-intercept form, how to calculate the slope, and how to find the y-intercept.

Step-by-Step Guide to Finding the Line Equation

To find the equation of a line passing through the points (2, -1) and (3, 2), we will follow these steps:

Step 1: Calculate the Slope (m)

The slope of the line is a measure of its steepness and is calculated using the formula:
m frac{y_2 - y_1}{x_2 - x_1}

Given the points (2, -1) and (3, 2), we can substitute the coordinates into the formula.

For point 1, (x_1 2) and (y_1 -1); for point 2, (x_2 3) and (y_2 2).

Plugging these values into the formula:

m frac{2 - (-1)}{3 - 2} frac{2 1}{1} frac{3}{1} 3

The slope (m) of the line is 3.

Step 2: Use the Point-Slope Form to Find the Equation

The point-slope form of a line is (y mx b). We already know the slope (m 3), and we will use one of the given points to find the y-intercept (b).

Let's use the point (2, -1) and substitute (m 3) and the point into the equation:

-1 3(2) b

-1 6 b

Solving for (b):
b -1 - 6 -7

Step 3: Write the Equation of the Line

Now we can write the equation of the line:

y 3x - 7

This is the equation of the line passing through the points (2, -1) and (3, 2).

Alternative Methods for Finding the Equation of the Line

There are other ways to find the equation of a line besides the point-slope form. Here, we will explore the intercept form and the general form of the line equation.

Intercept Form of a Line

The intercept form of a line is (y mx b). We have already determined that the slope (m 3) and the y-intercept (b -7), resulting in:

y 3x - 7

General Form of a Line

The general form of a line equation is (Ax By C 0). To convert the line equation from intercept form to the general form, we can rearrange the equation:

y 3x - 7

Subtract 3x from both sides to get:

-3x y 7 0

Multiplying both sides by -1 to make the coefficient of (x) positive:

3x - y - 7 0

Therefore, the line equation (y 3x - 7) can be written in the general form as:

3x - y - 7 0

Conclusion

In this article, you have learned how to find the equation of a line through two given points (2, -1) and (3, 2). The process involves calculating the slope, using the point-slope form, and finding the y-intercept. Additionally, you have explored alternative forms of the line equation, including intercept form and the general form. This knowledge will be valuable for further studies in mathematics and real-world applications involving linear relationships.

References

1. Math is Fun - Point-Slope Form 2. Free Math Lessons - Linear Equations