Question

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

Answer 1

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.

Answer 2

The way to solve any equation that contains however complex expressions containing a variable `X` on both sides is to transform it to a form `X=A`, where `A` is a known constant.

Sometimes it's impossible to accomplish. For instance, equation
`X+1 = X+2`
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
`X^2+1=5`
has two solutions: `X_1=2` and `X_2=-2`.

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:
`A*X+B=C*X+D`
where `A, B, C, D` are known constants and `X` is an unknown variable we have to find a value for.

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 `-C*X` to both (equal!) sides of an original equation. The result will be
`A*X+B-C*X=C*X+D-C*X`

The left side can be re-grouped (using commutative law of addition and subtraction), resulting in
`A*X-C*X+B`
The right side can be regrouped (using the same commutative law), resulting in
`C*X-C*X+D`
And both new expressions are equal as a result of these transformations:
`A*X-C*X+B=C*X-C*X+D`
Using the distributive law of multiplication relative to addition we can transform the left side into
`(A-C)*X+B`
Cancelling `C*X` and `-C*X` on the right results in `D`:
`(A-C)*X+B=D`

The equation now contains the unknown `X` only on the left.
Next step is add `-B` to both sides. On the left `B` and `-B` will cancel each other, resulting in the equation
`(A-C)*X=D-B`

Assuming `A-C` is not equal to zero, we can use another obvious rule of transformation:
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 `A-C`:
`X=(D-B)/(A-C)`
This is a solution.

Separately let's consider a case when `A-C=0`. In this case our equation after the transformations listed above will be
`0*X=D-B`
If constants `D` and `B` are equal to each other, any value of `X` will satisfy this and the original equation. We have infinite number of solutions.
If `D` and `B` are not equal to each other, no value of `X` would satisfy the equation - no solutions.

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