The balance method

Do the same to both sides to get x on its own, then check your answer in the first equation.

Engineers, nurses working out doses and anyone splitting a bill rearrange equations this way: keep it balanced, and every step stays true.

Backtracking stops working when $x$ is on both sides, as in $x + 7 = 3x + 1$ . The balance method works for every linear equation: whatever you do to one side, do to the other, so the scale stays level.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

One move on both sides

The left side has two bags of $x$ and 3 blocks. The right side has 11 blocks. Picture taking 3 blocks from each side.

2x + 3 = 11

. You take 3 from **both** sides. What equation is left?

2x = 8

2x = 11

2x + 3 = 8

x = 8

TryTry it on the board and compare the result with your prediction.

Keep it balanced

Use Take away and Share on the board. Each step does the same to both sides, so the scale stays level, and the list underneath writes your steps as algebra.

Try it yourselfYour edits are saved in this browser.

Get $x$ on its own on one side.

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Write it as algebra

Each line keeps the equation balanced.

Worked exampleBefore each step, say what you think comes next.

$3x + 5 = 20$ Start with the equation.
$3x + 5 - 5 = 20 - 5$ Take 5 from both sides.
$3x = 15$ Now three lots of $x$ are 15.
$\frac{3x}{3} = \frac{15}{3}$ Divide both sides by 3.
$x = 5$ $x$ is on its own.
$3(5) + 5 = 20 \checkmark$ Check in the first equation, not the last line.

Leo's check failed. Look at what he did to each side.

Either step keeps the balance. One keeps the numbers simpler.

Put the steps for solving $4x + 3 = 23$ in order.

Pieces (click to add)

Your program

Leo solves $2x + 3 = 11$ , but his check fails. Click the first line that goes wrong.

To solve

3x + 5 = 20

, which first step keeps whole numbers?Divide both sides by 3Take 5 from both sides

ModifyChange the problem on purpose and see what happens.

Harder scales

Each task loads a new scale. Same moves, bigger numbers.

Now there are bags of $x$ on both sides. Backtracking cannot solve this; the balance method can.

Need a hint?

Bags of x can be taken away too.

Another hint

Take 2 bags of x from both sides.

Bags and blocks on both sides. Get the bags on one side first.

Need a hint?

There are bags and blocks on both sides.

Another hint

Take one bag of x from both sides first.

Try it yourselfYour edits are saved in this browser.

Solve $4x + 7 = 31$ on the scale.

Use the board. This ticks when you get there.

Solve $5x = 2x + 12$ .

Use the board. This ticks when you get there.

Solve $x + 7 = 3x + 1$ .

Use the board. This ticks when you get there.

PractiseA few on your own, to check you have it.

Solve and check

Solve $5x + 4 = 39$ .

Solve $9x = 4x + 35$ .

Solve $6x + 2 = 4x + 12$ .

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Your own balance

Give it to a friend to solve with the balance method, then check their answer in your first equation.

Need a hint?

Pick two expressions with x, such as 4x + 1 and 2x + 7.

Another hint

Work out both when x = 3: 13 and 13. They match, so 4x + 1 = 2x + 7 works.

Write an equation with $x$ on both sides whose solution is $x = 3$ .

For teachers

Students solve linear equations by doing the same to both sides (the balance method), including x on both sides, and verify every answer in the first equation (AC9M7A03, node 3). About 50 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can solve an equation by doing the same thing to both sides, and show each step.
I can solve an equation with x on both sides.
I can check my answer by putting it back into the first equation, every time.

Before class

This is a hinge lesson: later modelling work (ratios, rates, money problems) depends on fluent solving and the checking habit. A student who skips the check will not notice a wrong answer.

Do the "x on both sides" scale yourself. The key move is taking a bag of x from both sides, which students rarely think of until they see the scale.

Running the lesson

Predict: each wrong option is a real slip (the right side left alone, the bags dropped, the 3 taken from the wrong side). Ask students who chose differently to explain.
Run: the scale never tips, because every button does the same to both sides. The list underneath is the written method: point at it.
Investigate: reveal the worked example one line at a time. Stress the last line: check in the first equation, not the line above.
Leo's mistake (one side only) is the misconception this lesson exists to fix. Ask: what happened to the right side?
Modify: for x + 7 = 3x + 1, let students struggle briefly, then ask what else they could take away from both sides.
Practise: students should write each step as a line of algebra, then check in the first equation. Equations with subtraction (adding to both sides) come in a later lesson.

When students get stuck

A student changes one side only (2x + 3 = 11 becomes 2x = 11). Ask: What did you do to the left side? Did the right side get the same? Fix: Write the step on both sides: 2x + 3 − 3 = 11 − 3.
A student checks in the last line (x = 4 → 4 = 4) and calls it verified. Ask: Which equation were we asked to solve? Fix: Check in the first equation: 2 × 4 + 3 = 11.
With x on both sides, the student is stuck. Ask: You took blocks from both sides. What else is on both sides? Fix: Take the smaller number of x from both sides first: x + 7 = 3x + 1 becomes 7 = 2x + 1.

Stretch and support

Stretch: 'I think of a number, double it and add 5 to get 19.' Write the equation, solve it with the balance method, and check. Then invent one where the answer must be a whole number of people, and explain why.

Support: Use the board for the Practise questions first, then write each button press as a line of algebra.

Where it leads

Next: equations from word problems, then graphing linear relations in Year 8, where the solution is where two lines meet.

Where it leads in class: rearranging formulas in science (speed = distance ÷ time), and money problems with a fixed fee plus a rate.

What each step is for

one move on both sides: Committing to the equation that one move leaves makes the move itself the thing students test on the board.
keep it balanced: Taking the same from both sides, then sharing both sides equally, keeps the equation true at every step and leaves x alone.
write it as algebra: Each move on the scale is a line of algebra, and checking in the first equation catches mistakes.
harder scales: Taking away bags of x from both sides is the same move as taking away blocks, and it handles x on both sides.
solve and check: Solving and then checking in the first equation becomes one routine.
your own balance: Writing an equation with x on both sides from a chosen solution shows the balance method is reversible.

Now you can

Solve an equation by doing the same thing to both sides, and show each step.
Solve an equation with x on both sides.
Check my answer by putting it back into the first equation, every time.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson10. Expanding bracketsThe number outside a bracket multiplies every term inside it. 3(2x + 4) is 3 rows of 2x + 4, so 6x + 12. Then collect any like terms.

Source: content/maths/algebra/balance-method/lesson.md