How to Find the Equation of a Line Given a Point and a Slope

Understanding the equation of a line is essential for anyone working with basic geometry or algebra. In this article, we will discuss how to derive the equation of a line when given a point it passes through and its slope. We will explore both the slope-intercept form and the point-slope form, using a specific example to illustrate the process.

The Problem

Suppose we are given a point and a slope for a line. The point is ((-2, -1)), and the slope is (frac{1}{2}). We want to find the equation of this line.

Step-by-Step Solution

Slope-Intercept Form: (y mx b)

The slope-intercept form of a line is [y mx b]. Here, (m) is the slope, and (b) is the y-intercept.

[y frac{1}{2}x b]

To find the y-intercept (b), we use the fact that the line passes through the point ((-2, -1)).

[-1 frac{1}{2}(-2) b]

Now, solve for (b):

[-1 -1 b]
[b 0]

Substituting (b) back into the equation, we get:

[y frac{1}{2}x]

Point-Slope Form: (y - y_0 m(x - x_0))

The point-slope form of a line is [y - y_0 m(x - x_0)]. Here, (m) is the slope, and ((x_0, y_0)) is a point on the line.

Substitute the given slope (m frac{1}{2}) and the point ((-2, -1)) into the formula:

[y - (-1) frac{1}{2}(x - (-2))]
[y 1 frac{1}{2}(x 2)]
[y 1 frac{1}{2}x 1]
[y frac{1}{2}x]

Conclusion

In both forms, we arrive at the same equation: (y frac{1}{2}x). This line passes through the point ((-2, -1)) and has a slope of (frac{1}{2}).

Additional Insights and Applications

Understanding how to derive the equation of a line is not only useful for solving mathematical problems but also has practical applications in various fields such as physics, engineering, and data science. For example, in physics, the slope of a line can represent the rate of change of a variable, such as velocity or acceleration.

To further explore the concept of line equations, you can also consider:

Using other points on the line to verify the equation. Deriving parallel and perpendicular lines. Exploring the relationship between two lines (intersecting, parallel, or perpendicular).

Remember, the slope and point-slope forms are powerful tools that simplify complex problems in mathematics and real-world scenarios.