How to Solve Algebra Equations
A free, hands-on way for primary and high-school students to learn algebra — drag a term across the equals sign and watch its sign flip.
Try it: drag terms across the equals sign
Drag a term across the equals sign and its sign flips. Combine like terms, expand brackets, then divide both sides to get x on its own.
What is an equation?
An equation is a maths sentence with an equals sign in the middle. It says that whatever is on the left is worth exactly the same as whatever is on the right. Solving an equation means working out the one value of the mystery number that keeps both sides equal.
An equation balances: both sides are equal
Think of the equals sign as the middle of a see-saw. The left side and the right side weigh the same — that's what "equals" means. In the tool above, the equation sits across the middle, with tiles on each side. As long as you do the same thing to both sides, the see-saw stays level and the equation stays true.
The variable: solving for
The letter — usually — is called the variable. It stands for a number you don't know yet. "Solving for " just means rearranging the equation until is sitting on its own on one side, with its value on the other: = 4. That's the finish line every time.
The golden rule: do the same to both sides
There's one rule behind everything: whatever you do to one side, you must do to the other. Take 5 off the left, take 5 off the right. Divide the left by 2, divide the right by 2. Keep the see-saw balanced and you can never go wrong.
How to solve a linear equation
A linear equation has the variable to the power of 1 — just , never squared. Solving one is a tidy-up job: move the plain numbers away from , gather the terms together, then divide to find what one is worth.
Move a term across the equals sign (change side, change sign)
Here's the move the tool is built around. To get a term off one side, drag it across the equals sign — and as it crosses, its sign flips. A +5 becomes a −5; a −3 becomes a +3. That's not a trick; it's the shortcut for "subtract 5 from both sides" done in one step. Maths teachers call it change side, change sign.
Example: Drag the +5 across the =. It lands as −5 on the right: 3 = 11 − 5 = 6.
Combining like terms
Like terms are terms with the same letter part — 3 and 2 are like terms because both are "lots of ". You add them by adding the numbers in front: 3 + 2 = 5 (three 's plus two more 's make five 's). But 3 and 2 are not like terms — one is lots of , the other is just a number — so they can't be joined. In the tool, drop one like tile onto another and it shows you the sum.
Example: 3 + 2 = 5, but 3 + 2 stays as 3 + 2 — you can't add a number of 's to a plain number.
Dividing both sides to get on its own
Once you reach something like 4 = 12, you're one step away. 4 means "4 lots of ", so to find one you share both sides into 4 equal parts — you divide both sides by 4. In the tool, drag the 4 underneath to make a fraction, then simplify: 12 ÷ 4 = 3, so = 3.
Equations with brackets
Expanding brackets: the distributive law
When a number sits just outside a bracket, it multiplies everything inside — not just the first thing. 2( + 3) means 2 × and 2 × 3, which is 2 + 6. This is called the distributive law. The commonest slip is multiplying only the first term and writing 2 + 3 — the tool's expand button shows every term being multiplied so the pattern sticks.
Example: 2( + 3) = 2 + 6; −2( − 1) = −2 + 2 (watch the signs).
Expand first, then solve
With a bracket in the way, expand it first, then solve the normal way. 2( + 3) = 14 becomes 2 + 6 = 14; move the +6 across to get 2 = 8; divide by 2 to get = 4. Bracket gone, problem solved.
Equations with on both sides
Gather the 's on one side
Sometimes appears on both sides: 5 + 2 = 3 + 8. Use the same move — drag an term across the equals sign and flip its sign — to collect all the 's on one side. Move the 3 across: 5 − 3 = 2, leaving 2 + 2 = 8. Then finish as before: 2 = 6, so = 3.
Example: 5 + 2 = 3 + 8 → 2 + 2 = 8 → 2 = 6 → = 3.
Why this matters on scholarship tests and Year 7–10 maths
ACER, Edutest and AAS scholarship maths sections lean on quick, accurate equation-solving — often dressed up as a worded problem. The Australian curriculum formally introduces solving linear equations in Year 7, with brackets and variables-on-both-sides following in Years 8–9. The faster the "change side, change sign" move becomes automatic, the more time you have for the hard questions.
Worked example: solve 2( + 3) = 14
Follow the same order every time: expand brackets, move terms across, then divide last.
- 1
Read the equation
We have 2( + 3) = 14. The 2 is multiplying everything inside the bracket, and we want to get on its own.
- 2
Expand the bracket
Multiply both terms inside by 2: 2 × = 2 and 2 × 3 = 6. The equation becomes 2 + 6 = 14.
- 3
Move the +6 across the equals sign
As it crosses, +6 becomes −6: 2 = 14 − 6, which is 2 = 8. (Same as subtracting 6 from both sides.)
- 4
Divide both sides by 2
2 means 2 lots of , so share both sides into 2: = 8 ÷ 2.
- 5
Simplify and check
8 ÷ 2 = 4, so = 4. Check: put it back — 2(4 + 3) = 2 × 7 = 14. Key takeaway: expand brackets first, then move terms across (flipping signs), then divide last.
Common mistakes to avoid
Moving a term across the = but forgetting to flip its sign
The single most common error, and the exact misconception the widget is built to fix. Every term that crosses the equals sign changes sign — + becomes −, − becomes +. The drag animation flips it for you so the rule sticks.
Adding unlike terms (thinking 3 + 2 = 5)
You can only combine terms with the same letter part. 3 + 2 = 5, but 3 + 2 stays as it is. The tool refuses to merge a number tile with an tile and tells you why.
Only doing it to one side
Dividing the left by 2 but not the right, or expanding a bracket but forgetting a term. The golden rule: whatever you do to one side, do to the other; when expanding 2( + 3), multiply every term inside.
Where algebra goes next
Solving equations is the doorway into the rest of algebra. From here the same skills lead into straight-line graphs (y = mx + c), function rules, and simultaneous equations (two equations solved together).
For curriculum revision, see our Year 6 maths guide (where number sentences and the idea of an unknown begin), browse the full Australian Curriculum index, or start a free practice test to try the kinds of algebra questions that come up in school assessments and scholarship exams.
Algebra vocabulary
The words you'll meet when you start solving equations.
- Equation
- A maths sentence saying two things are equal, with an = in the middle.
- Expression
- A bit of maths with no equals sign, like 3 + 5.
- Variable
- A letter (usually ) standing for a number you don't know yet.
- Term
- A single piece of an expression, like 3 or 5, separated by + or −.
- Coefficient
- The number in front of a variable — the 3 in 3.
- Constant
- A plain number on its own, like 5.
- Like terms
- Terms with the same letter part (3 and 2) that can be added together.
- Linear equation
- An equation where the variable is only ever to the power of 1 (just ).
- Inverse operation
- The opposite move that undoes another — subtract undoes add, divide undoes multiply.
- Expand
- Multiply out a bracket so 2( + 3) becomes 2 + 6.
- Distributive law
- The rule that the number outside a bracket multiplies everything inside.
- Solution
- The value of the variable that makes the equation true (e.g. = 4).
Frequently asked questions
What is an equation?
An equation is a maths sentence with an equals sign, saying the left side is worth the same as the right. Solving it means finding the one value of the variable (usually ) that keeps both sides equal — like = 4.
How do you solve for ?
Get on its own. Move plain numbers to the other side (flipping their sign as they cross the equals sign), combine like terms, then divide both sides by the number in front of . Whatever you do to one side, do to the other.
What does "move a term across the equals sign" mean?
It's a shortcut for doing the same thing to both sides. When a term crosses the =, its sign flips: +5 becomes −5. Teachers call it "change side, change sign". So 3 + 5 = 11 becomes 3 = 11 − 5.
How do you combine like terms?
Add the numbers in front of terms that share the same letter. 3 + 2 = 5 because three 's plus two 's make five 's. You can't combine 3 and 2 — one is lots of , the other is just a number.
How do you expand brackets?
Multiply everything inside the bracket by the number outside. 2( + 3) becomes 2 × + 2 × 3 = 2 + 6. The most common slip is multiplying only the first term — every term inside gets multiplied.
How do you solve equations with on both sides?
Drag the smaller term across the equals sign (flipping its sign) so all the 's are on one side. 5 + 2 = 3 + 8 becomes 2 + 2 = 8, then 2 = 6, so = 3. Then finish as normal.
Is algebra on the ACER, Edutest, or AAS scholarship test?
Yes — at the Year 6 and Year 7 entry level, the maths or numerical-reasoning section includes equation-solving and finding an unknown, often inside a worded problem. Quick, accurate solving frees up time for the harder questions.
What year do you learn to solve equations in the Australian curriculum?
Number sentences with an unknown start in primary (around Year 6). Solving linear equations formally arrives in Year 7 (AC9M7A04), with brackets and variables on both sides following in Years 8–9 — the levels this tool covers.