Using y = mx + c

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, $(x_1,y_1)$ and $(x_2,y_2)$, the gradient can be calculated using the formula:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

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 $(x_1,y_1)$ 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, $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-2}{3-1}=\frac{4}{2}=2$$

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 $(x_1,y_1)=(1,2)$: $$2=2(1)+c$$ $$2=2+c$$ $$c=0$$

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.