Finding the Linear Equation from Two Points

In the realm of mathematics, linear equations play a fundamental role in representing relationships between variables. A linear equation is an equation that can be written in the form y = mx + b, where m represents the slope and b represents the y-intercept. Understanding how to find the equation of a line given two points is a crucial skill in algebra and has applications in various fields, including physics, engineering, and economics.

Understanding the Slope

The slope of a line is a measure of its steepness. It represents the rate of change of the y-coordinate with respect to the x-coordinate. The slope is calculated as the ratio of the change in y (vertical change) to the change in x (horizontal change). We can write this mathematically as:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two given points.

Finding the Y-intercept

The y-intercept is the point where the line crosses the y-axis. It represents the value of y when x is equal to 0. To find the y-intercept, we can use the slope-intercept form of the linear equation, y = mx + b. We can substitute one of the given points (x1, y1) and the calculated slope m into this equation and solve for b:

y1 = m * x1 + b

b = y1 - m * x1

Putting it All Together

Once we have calculated the slope m and the y-intercept b, we can write the equation of the line in slope-intercept form:

y = mx + b

Example

Let's say we are given the points (2, 5) and (4, 11). We can find the equation of the line that passes through these points using the steps outlined above:

1. Calculate the Slope

m = (11 - 5) / (4 - 2)

m = 6 / 2

m = 3

2. Find the Y-intercept

We can use the point (2, 5) and the calculated slope m = 3 to find the y-intercept:

b = 5 - 3 * 2

b = 5 - 6

b = -1

3. Write the Equation

Now that we have the slope m = 3 and the y-intercept b = -1, we can write the equation of the line in slope-intercept form:

y = 3x - 1

Conclusion

Finding the linear equation from two points is a fundamental skill in algebra with wide-ranging applications. By understanding the concepts of slope and y-intercept, and following the steps outlined above, we can confidently derive the equation of a line that passes through any two given points.