Linear Equations Explained Simply: The Balance Method

Callum was in Year 9, and the linear equation 3x + 5 = 20 had already beaten him twice that week. At my kitchen table in Salford he wrote 3x = 20 + 5, then x = 25/3, and looked rather pleased. I asked why the 5 had become a plus. "Because it moves over the equals sign." And what happens to a number when it moves? "It changes sign." He stopped. "Oh. Hang on." He rubbed it out, wrote 20 − 5, got 15, then 5. Thirty seconds, no help from me.

Callum knew the rule by heart and played it like a ritual, with no idea what the ritual protects. Most of what makes this topic hard sits right there.

Linear equations explained simply: an equation says two sides weigh the same, like a balance sitting level. To solve it, you find the unknown (usually x) by doing the same thing to both sides and undoing the operations in reverse order until x stands alone. Then you check by putting your answer back into the original equation.

In England this lands in Key Stage 3, mostly Years 8 and 9. In the US it spreads from 6th to 8th grade. Either way, it only has to be understood once.

Linear equations explained simply: the equals sign is the first trap

From their first sums, children read 3 + 4 = 7 from left to right: a calculation on the left, the answer after the sign. The equals sign becomes a button marked "give me the result". Then 3x + 5 = 20 turns up, with numbers on both sides, and the button stops working.

An equals sign read as an "answer" button

Eric Knuth, Ana Stephens, Nicole McNeil and Martha Alibali (2006, Journal for Research in Mathematics Education) asked 177 middle school students in grades 6 to 8 (roughly ages 11 to 14) what the equals sign means. Only a minority gave a relational answer along the lines of "both sides are the same amount". Plenty said "add the numbers" or "the answer". The students with the relational view did better on equations, and that link held even after the researchers allowed for general maths ability using standardised test scores. The record is on ERIC.

McNeil and Alibali (2005, Child Development) found something close. Children aged 7 to 11 who clung to arithmetic habits, such as doing every operation to every number or reading the equals sign as "the total", learned less from a lesson on equations. In a second experiment, undergraduates who had those habits switched on beforehand did worse on equations too: doi.org/10.1111/j.1467-8624.2005.00884.x.

Both are single studies, so I wouldn't call either a law. They do match what I see on scripts.

Your child has already solved one

Year 1 pupils in England meet missing-number problems such as 7 = □ − 9, and American 1st graders find the unknown in 8 + ? = 11. The empty box is x in a costume. A child who writes 4 in the box of 3 + □ = 7 has solved a linear equation without a letter or a word of vocabulary. What changes in Year 8 is the notation: a letter instead of a box, and operations piled up around it.

Solving linear equations with the balance method

Picture a balance. On the left pan sit three identical opaque bags and 5 counters. On the right pan, 20 counters. It rests level. Take 5 counters off each pan and it still rests level: three bags against 15 counters. Share the 15 counters between the 3 bags and each bag holds 5. That is 3x + 5 = 20, and x = 5.

Four moves cover almost everything at this level:

1. Read the equation as a balance. Two sides, same weight, nothing more.
2. Undo in reverse order. You put socks on before shoes, so the shoes come off first. In 3x + 5, x was multiplied by 3 and then 5 was added. Take away 5, then divide by 3. These are the inverse operations.
3. Collect the x terms on one side whenever x appears on both.
4. Check. Put the answer back into the original equation. Every time, even when you feel sure.

Three examples, easiest first

Example 1: x + 7 = 12. One bag and 7 counters against 12 counters. Take 7 off each side: x + 7 − 7 = 12 − 7, so x = 5. Check: 5 + 7 = 12.

Example 2: 2x − 9 = 7. There is a minus sign in plain view, and the first move is still to add, because you undo whatever you can see. Add 9 to both sides: 2x = 16. Divide both sides by 2: x = 8. Check: 2 × 8 − 9 = 16 − 9 = 7.

Example 3: two bike-hire deals. Deal A costs £10 to start plus £4 an hour. Deal B costs £6 an hour with no starting fee. After how many hours do they cost the same? Write 4x + 10 = 6x. The x terms sit on both sides, so take 4x off each pan: 10 = 2x, so x = 5. Check: A gives 10 + 4 × 5 = £30 and B gives 6 × 5 = £30. Before 5 hours B is cheaper (at 3 hours, £18 against £22), and after 5 hours A wins. Swap pounds for dollars if you are in the US; the algebra doesn't care. A real question like this reaches a 13-year-old far better than a page of bare practice.

What to avoid: the shortcut taught first

My view, and I'll defend it: "change side, change sign" should never be the first thing a child learns about equations.

The shortcut is correct. It summarises what happens when you write − 5 on both sides without actually writing it. But the child who gets it first remembers the sentence and loses the reason, and the sentence then fires in the wrong places. A sign flipped for no reason here, a sign forgotten there. The mistakes I meet again and again:

• Flipping the sign of a number that stays on its own side.
• Doing something to the left and forgetting the right, so x + 7 = 12 turns into x = 12.
• Subtracting when the move was to divide: 2x = 16 giving x = 14.
• Skipping the check, which costs ten seconds.

A mum in Stockport wrote to me through EduBoost two winters ago about her daughter Zainab, in Year 8, who "knew her rule" and still dropped marks on every equations test. We went through three of her tests together. Sometimes she flipped a sign, sometimes she forgot to, with no pattern I could see. I took the rule away and added one habit: a vertical line to the right of each line of working, with the operation written beside it, | − 5 and then | ÷ 3. She grumbled ("that's extra writing") and kept going. A month later her mum told me the sign errors had gone. One girl proves nothing, but what she'd been doing before wasn't working.

My years as an examiner add a second reason. On a three-mark "solve" question, a script with clear working and a slip in the last line can still collect method marks. A bare wrong answer collects none. The margin note protects marks as well as understanding.

When the balance wobbles

Two kinds of equation trip up even children who got everything else. First the negative x: in −x = 4, the balance says the opposite of x is 4, so x = −4 (check: the opposite of −4 is 4). Then the divided x: in x/3 = 4, undo the division by multiplying both sides by 3, so x = 12. A child who really took in "undo in reverse order" needs no new rule for either.

Do I ban the shortcut? No. Once a child writes the margin note without thinking, they can skip the ink and keep the logic. That is the same move, minus the writing. How long that takes depends entirely on the child, and no chapter calendar will decide it.

Where linear equations sit in school (England and the US)

In England the Key Stage 3 programme of study expects pupils to solve linear equations in one variable, including forms that need rearranging, and the topic keeps returning through GCSE (DfE programme of study). Our GCSE maths key topic exercises show what the exam versions look like. In the US, Common Core spreads it over three grades: equations of the form x + p = q and px = q in 6th grade (6.EE.B.7), px + q = r in 7th (7.EE.B.4), and variables on both sides in 8th (8.EE.C.7), all listed under the Common Core 6.EE to 8.EE standards.

A Year 8 pupil and a 7th grader, both around 12 or 13, are on very similar ground.

What the research says

I keep coming back to Robert Siegler and colleagues (2012, Psychological Science), who analysed large, nationally representative datasets from the US and the UK. What children knew about fractions and division in primary school predicted their algebra and overall maths results in high school five or six years later, even after accounting for other maths skills, general intellectual ability, working memory, and family income and education: doi.org/10.1177/0956797612440101. That is correlation again. It is also why I wrote about fractions explained simply first. In my experience, a child who got through fractions on rules instead of pictures meets the same problem here, wearing a letter.

Year 8 equations are the first real gate between arithmetic and algebra. I'd rather children walked through it understanding than reciting.

Five-question quiz

1. Solve x + 9 = 21.
2. Solve 5x = 40.
3. Solve 3x + 4 = 19.
4. Solve 4x − 7 = 2x + 9.
5. A student writes 2x + 6 = 14, then 2x = 14 + 6 = 20, then x = 10. Where is the mistake, and what is the right answer?

Answers: 1) x = 12. 2) x = 8 (divide both sides by 5). 3) 3x = 15, so x = 5. 4) Take 2x off each side: 2x − 7 = 9, then 2x = 16, so x = 8; check: 4 × 8 − 7 = 25 and 2 × 8 + 9 = 25. 5) They added 6 where they should have taken it away: 2x = 14 − 6 = 8, so x = 4. Their answer fails the check, because 2 × 10 + 6 = 26, not 14.

If two answers wobble, bring the counters back out and redo the examples by hand before moving on to exercises. Ten minutes every two or three days does more than an hour the night before a test, which is the whole idea behind spaced repetition; our guide to spaced repetition has sample schedules. Another trick is to ask your child to say out loud what they do to both sides on every line, as in the Feynman technique. Anyone who can say it can usually do it.

For practice at their own pace, the EduBoost AI tutor builds harder variations and goes back over each mistake. Families in England can start with Year 9 maths tutoring, and families in the US with 8th grade math tutoring. If you'd rather have a person alongside, see our online tutoring options.

Back to Callum

Callum didn't get a new rule. I drew a balance on the back of an envelope, three bags and a handful of coins, and for three weeks I asked the same question at the end of every line: what did you do to both sides? By the third week he was writing | − 5 in the margin without being asked.

A child has learned linear equations when they can say what they are doing to both sides. Speed and shortcuts come later, and they last longer when they do. So the next time your child solves one, ignore the answer for a moment and ask only that.