Finding the Gradient and Equation of a Straight Line Passing Through Given Points

When working with straight lines in mathematics, one of the essential skills is to be able to calculate the gradient (or slope) and the equation of the line passing through two given points. In this article, we will demonstrate this process using two specific points: (-1, 2) and (8, 5).

Step 1: Finding the Gradient

The gradient (m) of a straight line that passes through two points (x1, y1) and (x2, y2) is given by the formula:

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

Using the points (-1, 2) and (8, 5), we can calculate the gradient as follows:

Let's substitute the coordinates into the formula:

x1, y1 -1, 2

x2, y2 8, 5

The calculation would be:

m (5 - 2) / (8 - (-1)) 3 / (8 1) 3 / 9 1 / 3

Therefore, the gradient of the straight line is 1/3.

Step 2: Finding the Equation of the Straight Line in Slope-Intercept Form

The equation of a straight line in slope-intercept form is given by:

y mx b

Where m is the gradient, and b is the y-intercept. We can use one of the points to find the y-intercept b. We will use the point (-1, 2) and substitute the gradient m 1/3 into the equation.

The equation is:

y (1/3)x b

Substituting the coordinates of the point (-1, 2) into the equation to find b:

2 (1/3)(-1) b

Solving for b:

2 -1/3 b

b 2 1/3 6/3 1/3 7/3

So, the equation in slope-intercept form is:

y (1/3)x (7/3)

Step 3: Converting the Equation to General Form

The general form of a line is given by Ax By C 0. To convert the slope-intercept form to general form, we need to eliminate the fraction and rearrange the equation:

y (1/3)x (7/3)

Multiplying the entire equation by 3 to clear the fraction:

3y x 7

Re-arranging all terms to one side of the equation:

-x 3y - 7 0

This can be rewritten as:

x - 3y - 7 0

Summary

The gradient of the straight line is 1/3.

The equation of the line in general form is x - 3y - 7 0 or equivalently, x - 3y 7 0.

Now you have a clear understanding of how to find the gradient and equation of a straight line passing through given points.

Additional Resources and Further Reading

For a deeper understanding, you might want to explore more resources on the topic, such as:

Google search for gradient calculator or slope calculator for additional tools. YouTube tutorials on straight lines and their equations for visual explanations. Math textbooks or online courses covering basic geometry and algebra.

These resources will help reinforce your understanding and provide practical examples for applying these concepts.