Understanding the Equation of a Line Through a Given Point with a Specific Y-Intercept

In the realm of mathematics, particularly in analytic geometry, the equation of a line can be found using several methods. One common approach is using the slope-intercept form, which allows us to determine the equation of a line given a point and the y-intercept. This method is particularly useful in various applications, including real-world scenarios and data analysis. In this article, we will explore the process of finding the equation of a line that passes through a given point and has a specified y-intercept.

Introduction to Slope-Intercept Form

The slope-intercept form of a line is a well-defined way of expressing the equation of a line. It is given by the formula:

y mx b

Where:

m represents the slope of the line. b represents the y-intercept of the line.

The Problem and Solution

Consider a scenario where we need to find the equation of a line that passes through a given point, (2, -5), and has a y-intercept of -3. Using the slope-intercept form, we can solve this problem step-by-step:

We know from the given information that the y-intercept b -3. To find the slope of the line, we use the formula for the slope: m frac{y_2 - y_1}{x_2 - x_1} In this case, we can use the point (0, -3) as one point and the point (2, -5) as the other point: x_1, y_1 (0, -3) and x_2, y_2 (2, -5) Substitute the values into the slope formula: m frac{-5 - (-3)}{2 - 0} frac{-2}{2} -1 Now that we have the slope m -1, we can write the equation of the line in slope-intercept form: y -1x - 3 or simply y -x - 3

Thus, the equation of the line is:

y -x - 3

Alternative Methods

There are multiple ways to find the equation of a line that passes through a given point and has a specified y-intercept. Here are a couple of alternative methods:

First, we determine the line's equation using the formula for a line passing through two points:

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

Using the points (0, -3) and (2, -5), we can write:

frac{y - (-3)}{-5 - (-3)} frac{x - 0}{2 - 0}

This simplifies to:

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

Multiplying both sides by 2:

y 3 -x

Thus, we get:

x y 3 0

Another way to approach this problem is to find the gradient/slope and then use the point-slope form:

Given a gradient of -1 (calculated as frac{-5 - (-3)}{2 - 0} -1), we use the formula:

y mx c

Substituting the point (2, -5) into the formula:

-5 -1(2) c

Solving for c (y-intercept):

c -3

Thus, the equation is:

y -1x - 3 or simply y -x - 3

Conclusion

By understanding the slope-intercept form and the y-intercept, we can find the equation of a line that passes through a given point and has a specified y-intercept. This method is not only useful in academic settings but also in practical applications involving data analysis and graphing. Whether you choose the slope-intercept form, the two-point form, or the point-slope form, the end result will be the same: the correct equation of the line. Understanding these methods will help you solve similar problems with ease.