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

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.