Using y = mx + c

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Topic summary

The equation y=mx+c is the general form of the equation of a straight line, where:

  • m is the gradient (or slope) of the line, which represents how steep the line is.
  • c 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 y=mx+c:

The equation y=mx+c describes a straight line. The gradient m is calculated by determining how much y changes for a given change in x, and c represents the value of y when x=0, i.e., where the line crosses the y-axis.

2. Finding the gradient m from two points:

If you are given two points on the line, (x1,y1) and (x2,y2), the gradient can be calculated using the formula:

m=y2y1x2x1

This formula calculates the "rise over run" or the change in y divided by the change in x, i.e., the vertical change divided by the horizontal change between the two points.

3. Finding the equation of the line:

Once the gradient m is known, you can substitute it into the equation y=mx+c. To find c, you can substitute the coordinates of one of the points into the equation and solve for c. The general steps are:

  • Find the gradient m using the formula above.
  • Substitute m into the equation y=mx+c.
  • Substitute the coordinates of one point (x1,y1) into the equation to solve for c.
  • Write the equation of the line in the form y=mx+c.

4. Example: Finding the equation of a line using two points:

Let’s find the equation of a line that passes through the points (1,2) and (3,6).

Step 1: Calculate the gradient m:

Using the formula for the gradient, m=y2y1x2x1=6231=42=2

Step 2: Substitute m=2 into the equation y=mx+c:

The equation becomes: y=2x+c

Step 3: Solve for c:

Now, substitute one of the points into the equation. Let's use (x1,y1)=(1,2): 2=2(1)+c 2=2+c c=0

Step 4: Write the equation of the line:

Now that we know m=2 and c=0, the equation of the line is: y=2x

5. Check the solution:

To verify, we can substitute the second point (3,6) into the equation y=2x: y=2(3)=6 This is correct, so the equation of the line is y=2x.

6. Special cases to note:

  • If the two points have the same x-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 y-coordinate, the line is horizontal, and the gradient is zero.

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