Introduction to Finding the Equation of a Line Through Two Points

To find the equation of a line passing through two given points, we follow a structured approach. This article will guide you through the process with several examples and clearly explain the method using various mathematical techniques.

Understanding the Problem

Suppose we have two points, (2, 3) and (4, 5), and we need to determine the equation of the line that passes through these points. This can be done by calculating the slope of the line and then using the point-slope form of the equation of a line.

Step 1: Calculate the Slope (m)

The slope of a line is a measure of its steepness and can be calculated using the formula:

[ m frac{y_2 - y_1}{x_2 - x_1} ]

Here, (x_1, y_1) (2, 3) and (x_2, y_2) (4, 5).

Calculation:

[ m frac{5 - 3}{4 - 2} frac{2}{2} 1 ]

Step 2: Use the Point-Slope Form of the Equation of a Line

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

[ y - y_1 m(x - x_1) ]

Using point (2, 3) and the slope m 1, we have:

[ y - 3 1(x - 2) ]

Let's simplify this equation further:

[ y - 3 x - 2 ]

[ y x - 1 ]

Therefore, the equation of the line through the points (2, 3) and (4, 5) is:

[ y x - 1 ]

Alternative Methods

There are multiple ways to derive the equation of a line through two points. Below are a couple of additional methods:

Method 1: Using the Two-Point Form

The two-point form of the equation of a line is given by:

[ frac{y - y_1}{y_2 - y_1} frac{x - x_1}{x_2 - x_1} ]

Substituting (x_1, y_1) (2, 3) and (x_2, y_2) (4, 5) into the above formula, we get:

[ frac{y - 3}{5 - 3} frac{x - 2}{4 - 2} ]

Simplifying, we have:

[ frac{y - 3}{2} frac{x - 2}{2} ]

[ y - 3 x - 2 ]

[ y - x -1 ]

[ y x - 1 ]

Method 2: Using Parametric Form

The parametric form of the line can be derived as:

[ xy left(frac{x_1 t(x_2 - x_1)}{y_1 t(y_2 - y_1)}right) left(frac{y_1 t(y_2 - y_1)}{x_1 t(x_2 - x_1)}right) ]

For our points (2, 3) and (4, 5), this simplifies to:

[ x 2 t(4 - 2) ]

[ y 3 t(5 - 3) ]

By solving for t in both equations, we find:

[ t frac{x - 2}{2} ]

[ t frac{y - 3}{2} ]

Equate the two expressions for t:

[ frac{x - 2}{2} frac{y - 3}{2} ]

[ x - 2 y - 3 ]

[ y x - 1 ]

Conclusion

Through various methods, we have derived the equation of the line passing through the points (2, 3) and (4, 5). The final equation is:

[ y x - 1 ]

Understanding these methods is crucial for solving similar problems in geometry and algebra. By mastering these techniques, you will be well-equipped to tackle more complex mathematical challenges.