Introduction to Linear Equations and Slope

A linear equation is a fundamental concept in algebra and geometry, representing a straight line on a coordinate plane. It establishes a relationship between the vertical position (y-axis) and the horizontal position (x-axis) of points. The general form of a linear equation is y mx b, where:

m: the slope, representing the rate of change of y with respect to x. b: the y-intercept, which is the value of y when x 0.

Understanding the Slope

The slope of a line, denoted as m, can be calculated using two points on the line. The formula for slope is:

Slope (m) frac{y_2 - y_1}{x_2 - x_1}

This formula represents the change in y-coordinates over the change in x-coordinates, often referred to as the rise over the run.

Finding the Y-Intercept (b)

To find the y-intercept, we can use the slope-intercept form of the equation and substitute x 0. Alternatively, we can use the two-point form of the equation to find b.

Example: Finding the Equation of a Line Given Two Points

Given two points (x_1, y_1) and (x_2, y_2), we can find the equation of the line in slope and y-intercept form using the following steps:

Calculate the slope m using the formula:

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

Substitute the slope and one of the points into the slope-intercept form: y mx b. Solve for b. Substitute the values of m and b into the equation to get the full equation of the line: y mx b.

Let's consider the points (x_1, y_1) and (x_2, y_2) to be A(x_1, y_1) and B(x_2, y_2). We can write the two equations:

y_1 mx_1 b y_2 mx_2 b

Subtract the first equation from the second:

y_2 - y_1 m(x_2 - x_1)

Solve for b by substituting m frac{y_2 - y_1}{x_2 - x_1} into one of the original equations:

b y_1 - m x_1 y_1 - frac{y_2 - y_1}{x_2 - x_1} x_1

The equation of the line in slope and y-intercept form is:

y frac{y_2 - y_1}{x_2 - x_1} x left(y_1 - frac{y_2 - y_1}{x_2 - x_1} x_1right)

Application to Two Points

Given two points A(x_1, y_1) and B(x_2, y_2), the equation of the line is:

y frac{y_2 - y_1}{x_2 - x_1} x left(y_1 - frac{y_2 - y_1}{x_2 - x_1} x_1right)

This is the final equation of the line in slope and y-intercept form, derived using the two given points.

Conclusion

Understanding the process of finding the slope and y-intercept from two points is crucial for solving problems involving linear equations. By mastering this technique, you can effectively determine the equation of any line given just two points.