Variables on Both Sides
Gather variable terms on one side and constants on the other.
About this topic
When the variable appears on both sides, such as 5x − 3 = 2x + 9, first gather all the variable terms on one side by adding or subtracting a term from both sides. Then solve the remaining equation as usual. Here 3x − 3 = 9, so x = 4.
Moving terms to keep the coefficient positive reduces sign mistakes. If the variables cancel and the remaining statement is false, there is no solution; if it is true, every number is a solution. This models situations like comparing two phone plans. For 4x + 1 = 4x + 5, subtracting 4x leaves 1 = 5, which is never true, so there is no solution. For 2(x + 1) = 2x + 2, every value works.
Suppose Plan A costs 10 dollars plus 0.05 per minute and Plan B costs 4 dollars plus 0.11 per minute. They cost the same when 10 + 0.05m = 4 + 0.11m. Subtract 0.05m and 4 from both sides to get 6 = 0.06m, so m = 100 minutes. Below 100 minutes Plan B is cheaper, and above it Plan A wins.
Students often move a term across without changing its sign, which is wrong. Remember you are adding or subtracting the same thing to both sides, not just moving it. Check by substituting the answer into both sides and making sure they agree.
Key ideas
- Subtract the smaller variable term
- Then solve a simple equation
- Same value both sides means identity
Quick check
Use cases: where Variables on Both Sides helps
- Homework: review the key idea behind Variables on Both Sides before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Variables on Both Sides without notes.
- Visual learning: play with the interactive lab to see how Variables on Both Sides changes when you adjust its inputs.
- Teaching a friend: explain Variables on Both Sides in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Variables on Both Sides?
Gather variable terms on one side and constants on the other.
What will I learn in Variables on Both Sides?
Subtract the smaller variable term. Then solve a simple equation. Same value both sides means identity.
Is Variables on Both Sides part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Variables on Both Sides lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Variables on Both Sides?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.