How to Find the Equation of a Straight Line with Given Gradient and Point

In this article, we will explore the process of finding the equation of a straight line given a specific gradient and a point through which it passes. This is useful in various mathematical and real-world applications, such as graphing, regression analysis, and solving geometric problems.

Understanding the Problem

Given a point and a gradient (or slope), we can determine the equation of the line that passes through the given point. The point we have is (2, 3), and the gradient is -2. Let's break down the steps needed to find the equation of the line.

The Point-Slope Form

The point-slope form of the equation of a line can be written as:

y - y_1 m(x - x_1)

Where x_1 and y_1 are the coordinates of the given point, and m is the gradient of the line.

Substituting the Given Values

Let's substitute the given values into the point-slope form:

y - 3 -2(x - 2)

Simplifying the Equation

To simplify this equation, we first distribute the -2:

y - 3 -2x 4

Next, we add 3 to both sides of the equation to solve for y:

y -2x 7

This is the equation of the straight line in slope-intercept form, which is y mx b, where m is the gradient and b is the y-intercept.

A General Approach

Alternatively, we can use the point-slope form more generally. Given a point (x_0, y_0) and a slope m, the point-slope equation is:

y - y_0 m(x - x_0)

In our example, x_0 2 and y_0 3, and the slope m -2. Plugging these values into the equation:

y - 3 -2(x - 2)

Distributing the -2 gives:

y - 3 -2x 4

Adding 3 to both sides:

y -2x 7

Conclusion

By using the point-slope form and substituting the given values, we have successfully found the equation of the straight line. The equation of the line in slope-intercept form is:

y -2x 7

This form is particularly useful as it directly provides the gradient and the y-intercept of the line. It can be converted to the standard form, which is Ax By C 0, for further analysis if needed:

2x y - 7 0

Understanding these steps and forms is crucial for solving a wide range of problems related to straight lines and their equations.