Understanding the Equation of a Line Given a Point and Slope

In this article, we will explore how to find the equation of a line given a point and its slope. This is a fundamental concept in algebra and is often used in various fields, including data analysis, engineering, and physics. Let's delve into the methods and step-by-step processes to solve such problems.

Introduction to Linear Equations and Slope

A linear equation in two variables can be written in the form y mx b, where m is the slope of the line and b is the y-intercept. When we need to find the equation of a line given a point and its slope, we use the point-slope form of a linear equation, which is y - y_1 m(x - x_1). Here, m is the slope, and (x_1, y_1) is a point on the line.

Solving for the Equation Given a Point and Slope

To find the equation of a line passing through the point (-2, -2) with a slope of 2, we can use the point-slope form. Let's walk through the process step by step.

Step 1: Substituting Values into the Equation

Given the point (-2, -2) and the slope m 2, substitute these values into the point-slope equation:

y - y_1 m(x - x_1)
y - (-2) 2(x - (-2))
y 2 2(x 2)

Step 2: Simplifying the Equation

Next, simplify the equation by distributing the slope (2) on the right side:

y 2 2(x 2)
y 2 2x 4

Step 3: Isolating the Variable y

Finally, subtract 2 from both sides to isolate y:

y 2 - 2 2x 4 - 2
y 2x 2

The equation of the line is: y 2x 2.

Alternative Methods and Verifications

Let's explore two alternative methods to verify the solution:

Method 1: Using Simple Algebra

Another way to reach the equation is by using simple algebraic manipulation. Given the equation:

y 2x 2

Let's test it with the given point (-2, -2). Substituting x -2:

y 2(-2) 2
y -4 2
y -2

This confirms that the point (-2, -2) lies on the line, verifying the equation.

Method 2: Using Substitution

Another approach is to use the form y mx - (mx - b). Given the point (-2, -2) and the slope m 2, substitute these values:

y 2x - (2(-2) - (-2))
y 2x - (-4 2)
y 2x 2

This confirms the same equation.

Conclusion

In this article, we have explored the process of finding the equation of a line given a point and its slope. The point-slope form of a linear equation is a powerful tool in algebra. Whether you use the point-slope form, simple algebra, or substitution, the result is the same: the equation of the line is y 2x 2.

Key Takeaways

The point-slope form of a linear equation is y - y_1 m(x - x_1). To verify the equation, substitute the given point into the equation. The slope of a line can be used to describe its steepness and direction.

Understanding these concepts will help you solve various problems involving linear equations in the future.