Solving a Linear Equation

Flash Tutorial

This flash animation demonstrates how to solve a simple linear equation. If you do not have flash or if you prefer a text based tutorial, keep reading!

What is a Linear Equation?

First, let's take a look at a few exmaples.

A linear equation is an equation in which the unknown variable has no exponents.
E.g. 4x+5=25 is a linear equation. Notice the variable "x" has no exponent.
Here is an example of a non-linear equation: 3x^2 + 3 = 30. Notice that the variable "x" has an exponent of 2.

The Steps to Solve a Linear Equation

There are three simple steps to solving any linear equation.

  1. Combine like terms
  2. Isolate the variable using addition and subtraction
  3. Isolate the variable using multiplication and division
  4. The equation is solved! (check your answer)
Let's take a look at these steps as we solve the following linear equation: Linear Problem

Combine like terms

When we say "combine like terms" we are referring to the terms in the equation that have the same variable (in this case "x"). To combine the 14x and 5x terms, we can do one of two things. We could either subtract 14x from both sides, or we could subtract 5x from both sides. I prefer subtracting the smaller number so that we are left with a positive coefficient on our x term. Let's do that now.

Step 1) linear problem 2 (subtract 5x from each side)
Step 2) linear problem 3 (group like terms so it is easier to simplify)
Step 3) linear problem 4 (simplify)

Isolate the variable using addition and subtraction

Next, we want to isolate the variable on one side of the equation using addition and subtraction. After combining like terms we are left with the equation linear problem 4. Our next goal is to add or subtract something from both sides of the equation so that the 9x term is left by itself. In this case, you might see that we can do this by adding 3 to each side of the equation.

Step 1) linear problem 5 (add 3 to each side)
Step 2) linear problem 6 (simplify)

Isolate the variable using multiplication and division

Finally, we will solve the equation by multiplying or dividing both sides of the equation by something so that we are left with x on one side of the equation and a number on the other. In this case, we will divide each side of the equation by 9.

Step 1) linear problem 7 (divide each side by 9)
Step 2) linear problem 8 (simplify)

The equation is solved!

After following three easy steps we have solved the equation. Now that we have a value for x, we can plug that value back into the original equation to check our answer. We should be able to put 2 in place of x in the original equation, evaluate both sides and end up with the same number on both sides. Let's check our answer!

Step 1) linear problem 1 (original equation)
Step 2) linear problem 9 (substitute 2 for x)
Step 3) linear problem 10 (following the order of operations simplify by multiplication first)
Step 4) linear problem 11 (finish simplifying using addition and subtraction

Since 25 = 25 we know that 2 is the correct value for x. Congratulations! You now know how to solve a linear equation. Try the quiz for this topic to test your new skill set.