Variables on Both Sides

What changes?

In earlier equations, the variable often ended up on one side:

Now the variable can appear on both sides:

That does not mean there are two different variables. Both x terms represent the same number. The job is to use equivalent moves until the equation tells you what that number is.

The big move is to collect the variable terms on one side. Usually that means subtracting the smaller variable term from both sides.

One solution

Solve:

Subtract 2x from both sides:

Add 12 to both sides:

Divide both sides by 2:

This equation has one solution. Check it:

Both sides are 16, so x = 7 satisfies the original equation.

When brackets appear first

Distribute before collecting variable terms:

Expand the left side:

Subtract 2x from both sides:

Add 3 to both sides:

Same solution, but the first move was different: the bracket had to be expanded before the variable terms could be collected.

Three possible endings

Variables on both sides can end in three different ways.

One solution: the variable does not disappear.

No solution: the variable disappears and leaves a false statement.

That can never be true, so there is no solution.

Infinite solutions: the variable disappears and leaves a true statement.

That is always true, so every value of x works.

Misconception probe

Where it shows up in real life

Comparing two fundraisers. One Grade 7 group already has $12 in the cash box and sells tickets at $4 each. Another group starts with $2 and sells the same kind of tickets at $2 each. When do the totals match?

This one has variables on both sides because both groups sell tickets. Subtract 2t from both sides, then compare the remaining totals:

That gives a negative ticket count, so the equality is not meaningful in this real context. The algebra still solves; the context tells you whether the solution makes sense.

Worksheet

For each equation, collect variable terms, simplify, then classify the ending.

Going further

The next lesson turns real situations into equations. Variables on both sides appear whenever two changing quantities are being compared: two plans, two rates, two fundraisers, or two costs. The algebra finds the possible match point; the context decides whether that match point makes sense.