# How do you solve multi step equations with variables on both sides?

##### 2 Answers

Collect all terms that involve the variable on one side and all terms that do not involve the variable on the other side. (by adding and/or subtracting.)

Then divide both sides by the coefficient of the variable.

The way to solve any equation that contains however complex expressions containing a variable

Sometimes it's impossible to accomplish. For instance, equation

cannot be transformed in this format, then we say that an equation has no solutions.

Sometimes the transformations lead to multiple solutions. For instance, equation

has two solutions:

The main question is, *how to transform a given equation to a form rendering a solution*. This should be addressed separately for different kinds of equations. Let's consider the simplest type - **linear equations**.

The linear equation that contains an unknown variable on both sides of an equation can be presented in the following general format:

where

Let's use the obvious rule of transformation:

*if there are two equal values and we add the same value to both of them, the result will be two equal values*.

In our case let's add a value **equal!**) sides of an original equation. The result will be

The left side can be re-grouped (using commutative law of addition and subtraction), resulting in

The right side can be regrouped (using the same commutative law), resulting in

And both new expressions are equal as a result of these transformations:

Using the distributive law of multiplication relative to addition we can transform the left side into

Cancelling

The equation now contains the unknown

Next step is add

Assuming

*if there are two equal values and we divide them by the same number not equal to zero, the result will two equal values*.

So, divide both sides by

This is a solution.

Separately let's consider a case when

If constants

If

This is a complete analysis of all possible cases for linear equations with unknown variable on both sides.