The equation is the general form of the equation of a straight line, where:
- is the gradient (or slope) of the line, which represents how steep the line is.
- is the y-intercept, which represents the point where the line crosses the y-axis.
In this guide, we will discuss how to use the equation to find the equation of a straight line and how to calculate the gradient when given two points on the line.
1. Understanding the equation :
The equation describes a straight line. The gradient is calculated by determining how much changes for a given change in , and represents the value of when , i.e., where the line crosses the y-axis.
2. Finding the gradient from two points:
If you are given two points on the line, and , the gradient can be calculated using the formula:
This formula calculates the "rise over run" or the change in divided by the change in , i.e., the vertical change divided by the horizontal change between the two points.
3. Finding the equation of the line:
Once the gradient is known, you can substitute it into the equation . To find , you can substitute the coordinates of one of the points into the equation and solve for . The general steps are:
- Find the gradient using the formula above.
- Substitute into the equation .
- Substitute the coordinates of one point into the equation to solve for .
- Write the equation of the line in the form .
4. Example: Finding the equation of a line using two points:
Let’s find the equation of a line that passes through the points and .
Step 1: Calculate the gradient :
Using the formula for the gradient,
Step 2: Substitute into the equation :
The equation becomes:
Step 3: Solve for :
Now, substitute one of the points into the equation. Let's use :
Step 4: Write the equation of the line:
Now that we know and , the equation of the line is:
5. Check the solution:
To verify, we can substitute the second point into the equation : This is correct, so the equation of the line is .
6. Special cases to note:
- If the two points have the same -coordinate, the line is vertical, and the gradient is undefined (because the denominator in the gradient formula is zero).
- If the two points have the same -coordinate, the line is horizontal, and the gradient is zero.