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

##### 2 Answers
Mar 26, 2015

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.

Mar 26, 2015

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

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

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
$\left(A - C\right) \cdot X = D - B$

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 $A - C$:
$X = \frac{D - B}{A - C}$
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
$0 \cdot X = D - B$
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.