Introduction to Linear Equations

This article aims to introduce and explain the method of finding the equation of a line given two points on the line. By utilizing the concepts from slope-intercept form and coordinate geometry, we can derive the line's equation step-by-step. The goal is to provide clarity and understanding, making it easier for readers to apply these concepts in various mathematical and real-world scenarios.

Understanding the Slope-Intercept Form of a Line

The slope-intercept form is a fundamental formula used to represent a linear equation. It is given by the equation:

y mx b

where m is the slope of the line and b is the y-intercept, the point where the line intersects the y-axis.

Step-by-Step Process to Find the Equation

Step 1: Calculate the Slope

The first step in finding the equation of the line is to calculate its slope. The formula for the slope is:

m (y? - y?) / (x? - x?)

Let's use the points (0, 3) and (5, -3) to demonstrate this process.

Step 2: Use the Y-Intercept

The y-intercept b can be found by substituting one of the given points into the slope-intercept form equation. Since the point (0, 3) is given, we can determine that:

b 3

Step 3: Write the Equation

Now we can substitute the calculated values of m and b into the slope-intercept form formula:

y -6/5x 3

This is the equation of the line that passes through the points (0, 3) and (5, -3).

The Point-Slope Form and Its Application

Another method to find the equation of a line is to use the point-slope form:

(y - y?) m(x - x?)

Using the point (0, 3) and the slope -6/5, we can write the equation as:

y - 3 -6/5(x - 0)

which simplifies to:

y -6/5x 3

Verification and Conversion to General Form

To verify, we substitute the given points back into the equation:

For (0, 3): 3 -6/5(0) 3

For (5, -3): -3 -6/5(5) 3

This confirms that the equation is correct.

Alternatively, we can convert the equation to the general form:

6x 5y - 15 0

This form can be useful for certain applications such as graphing or solving systems of equations.

Conclusion

In summary, to find the equation of a line given two points, we first calculate the slope using the formula m (y? - y?) / (x? - x?). Then, we use one of the points to find the y-intercept. Finally, we substitute these values into the slope-intercept form to get the equation of the line. This method can be applied to any pair of points on a line to find its equation.