Solving linear equations is a fundamental part of solving linear problems. Linear equations are defined as equations that have variables that are only multiplied by numbers, not divided by numbers or raised to higher powers.

For example, x = 2y is a linear equation, but x = 2y2 or x = 2y–1 are not. Linear equations can be solved in two main ways: by adding or subtracting the same value to both sides of the equation to make it look like the opposite operation, then solving for the new variable, or finding another variable that works as the opposite operation and solving for it.

Solving linear equations can be tricky when there are more than **one possible solution**. It is important to check that you only solved for one variable and did not solve for both!

There are many ways to **apply solving linear equations** in **real life situations**. One common way is using them to solve simple systems of linear equations.

## Add a negative number to both sides of the equation

Solving linear equations involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number. You can also rearrange the left side of the equation.

For example, if you have the equation 4b + 6 = 2 – B + 4, you can divide both sides by 2 and then add B to get 2 + B + 4 = 4b + 6. Now you can solve for b by substituting b for 4b.

Solving linear equations is useful because it allows you to find what value(s) fit inside the equation. For example, if you have the equation 2 + B + 4 = 4b + 6 and you solve for B, you get B = –2. Therefore, only –2 fits inside the equation 2 + B + 4 = 4b + 6.

## Divide both sides of the equation by a negative number

To solve an equation that has a negative number on the left side of the equal sign, you must divide both sides of the equation by the negative number. This process is called negation.

For example, let’s look at the equation 4b + 6 = 2 – B + 4. To solve this equation, you **would divide** both sides by -B.

The variable B represents a positive number, so dividing both sides by -B would simply mean switching the sign of the number.

Whether you switch the sign or not does not matter; it will still solve the equation correctly.

If you divided both sides by -B and then realized that did not change the value of the solution, then you knew that B was a positive number.

By solving this equation, you discovered that B is a positive number of 5.

## Take the inverse of both sides of the equation

In math, the inverse of a number or a variable is the value that, when combined with the original number or variable, *produces zero* as a result. For example, the inverse of 4 is −4, **since subtracting 4** from any **number produces zero**.

In linear equations, solving for one variable in terms of another variable can be done by using the inverse function. By doing this, you are solving for what is on the left side of the equals sign in terms of what is on the right side.

The solution to an equation can be found by **putting one variable** in terms of the other and then doing a little arithmetic to find out how much of one variable is needed to get zero as a result.

## Solve using algebraic techniques

Solving linear equations is the process of finding what value or values solve the equation. The solution can be a number, called a solution, or it can be a set of numbers, called a solution set.

There are **three main ways** to solve linear equations. These are substitution, elimination, and averaging. Each *method uses different strategies* to solve the equation, so try out all three to see which **one works best** for you!

Substitution is by far the easiest way to solve an equation if you know how to do it correctly. The trick is figuring out how to place one variable in place of the other in the original equation.

To solve an equation using substitution, first rewrite the equation so that one variable is on the left side of the equals sign and one is on the right side. Then, find a value for the unknown variable and replace it with its counterpart on the left or right side of the equals sign.

## Check your solution

Once you have your solution, check it by solving the original equation for the other variable. In this case, you *would solve 4b* + 6 = 2 – B + 4 for b.

If your solution variable was b and your solution was 2, then you **would check** by subtracting 2 from both sides of the original equation. You would then solve the new equation for B to see if it matched your solved for B in the solution of the original equation.

You **could also check** your solution by graphing the two solutions. Graphically comparing the two solutions will show you if your solution is correct or not. If both graphs match, then your solution is correct!

Whether or not your solution is correct, always check to make sure that it is not incorrect.