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 XX on both sides is to transform it to a form X=AX=A, where AA is a known constant.

Sometimes it's impossible to accomplish. For instance, equation
X+1 = X+2X+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=5X2+1=5
has two solutions: X_1=2X1=2 and X_2=-2X2=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+DAX+B=CX+D
where A, B, C, DA,B,C,D are known constants and XX 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*XCX to both (equal!) sides of an original equation. The result will be
A*X+B-C*X=C*X+D-C*XAX+BCX=CX+DCX

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

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

Assuming A-CAC 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-CAC:
X=(D-B)/(A-C)X=DBAC
This is a solution.

Separately let's consider a case when A-C=0AC=0. In this case our equation after the transformations listed above will be
0*X=D-B0X=DB
If constants DD and BB are equal to each other, any value of XX will satisfy this and the original equation. We have infinite number of solutions.
If DD and BB are not equal to each other, no value of XX would satisfy the equation - no solutions.

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