A root is where a line (or a curve) crosses the

We can find a root in two ways; one method is to find the root algebraically and the other method is to find a root graphically.

*x*axis. Linear lines only have one root (other equations such as quadratic and cubic functions can have more than one root).We can find a root in two ways; one method is to find the root algebraically and the other method is to find a root graphically.

**Finding the Root Algebraically**

A root is where the line intercepts the

*x*-axis. The

*y*coordinate for any point on the

*x*-axis is zero. Therefore, we can find the root of an equation algebraically by subbing in

*y*as zero into the equation.

**Example 1**

What is the root for the equation

*y*= 3

*x*+ 6.

We find the root for this equation by subbing in

*y*as 0. This gives us the equation:

We then find the value of

*x*by solving the equation. We do this by isolating all of the*x*’s; we get all of the*x*’s to one side of the equation and all of the numbers to the other side. There are more*x*’s on the right, so it makes sense to have*x*’s on the right and numbers on the left. Therefore, we move the 6 from the right side of the equation to the left, which we can do by taking 6 from both sides of the equation.We now have all of the

*x*’s on one side and all of the numbers on the other side. The next step is to divide both sides by the coefficient of*x*, which is 3 (we divide by 3 because we want to find the value of*x*and not 3*x*).Therefore, the root of the equation has an

The second method is to find the root graphically, which is where we plot the line and find the

*x*coordinate of -2. The*y*coordinate for the root was 0, so this means that the root has the coordinates (-2, 0).**Finding the Root Graphically**The second method is to find the root graphically, which is where we plot the line and find the

*x*coordinate for where the line crosses the*x*axis. The graph from example 1 is plotted below.From the above graph, we can see that the line crosses the

*x*-axis at (-2, 0). Therefore, the root of the equation is where*x*is -2.**Example 2**

The line that is plotted on the graph below has the equation

*y*= -6

*x*+ 24. What is the root of the line below?

The line intersects the

Let’s now find the root algebraically. We solve it algebraically by subbing in

*x*axis at (4,0). Therefore, the root of this equation is 4.Let’s now find the root algebraically. We solve it algebraically by subbing in

*y*as 0 into the equation of the line.The easiest way to find the value of

*x*is to get all of the*x*’s to the side of the equation that currently has more*x*’s. Currently, there are more*x*’s on the left side of the equation than the right; 0*x*is more*x*’s than -6*x*. Therefore, we are going to get all of the*x*terms to the left and all of the numbers to the right. We need to move the -6*x*from the right side of the equation to the left and we do this by doing the opposite; we add 6*x*to both sides.We then divide both sides by 6 (the coefficient of

*x*) because we want to find the value of*x*and not 6*x*.Therefore, the root for the equation is 4, which is the same answer as the graphical method.

Usually, you will not be given a diagram of the line on a graph, which means that you will be unable to spot the root straight away. When this is the case, it is best to find the root by subbing in

**Final Note**Usually, you will not be given a diagram of the line on a graph, which means that you will be unable to spot the root straight away. When this is the case, it is best to find the root by subbing in

*y*as 0 and solving to find the value of*x*. This is because the graphical method of plotting the line and finding where the line passes through the*x*axis will take considerably longer to find the root of the equation compared with finding the root algebraically.