SumNinja

Solve equations when x changes position

About 3 min

Work out what x stands for before choosing the step. A starting amount and an amount taken away play different roles.

A game starts you with 12 energy points. After a move, 5 remain. How much did the move use? In 12 − x = 5, x is the amount you used. In x − 7 = 5, x is the starting amount. The position changes the story.

The starting amount splits into used and left

Twelve energy points split into seven used and five left. Used plus left must equal the start.

Recover the energy used

Find x in 12 − x = 5.

  1. Write start as used plus left

    12−x+x=5+x12-x+x=5+x

    Add x to both sides. You get 12 = 5 + x: the start equals the points left plus the points used.

  2. Take away the part you know

    12−5=x⇒x=712-5=x\quad\Rightarrow\quad x=7

    Take 5 from both sides. Seven points went into the move.

  3. Check the original story

    12−7=512-7=5

    Using seven points from twelve leaves five. This check catches a wrong choice of operation.

The same role check helps with division

Now spend 12 points on three equal moves. Each move costs x points, so 12 ÷ x = 3. Multiply both sides by x: 12 = 3x. Divide by 3 to find x = 4.

In x ÷ 3 = 4, x means the full amount instead. Multiply both sides by 3 to find x = 12. You cannot divide by zero.

You still keep both sides equal. Reading the role of x helps you choose how. A rule such as “change every minus to plus” misses that choice.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Find the removed amount

18−x=1118-x=11

Which amount does x stand for? Find it and check the original equation.

Show the answerHide the answer
x=7x=7

Add x, then take 11 from both sides. This gives 18 = 11 + x, so x = 7.

Find the missing divisor

20÷x=520\div x=5

Twenty points pay for five equal moves. What does each move cost?

Show the answerHide the answer
x=4x=4

Multiply by x to get 20 = 5x. Divide by 5. Then check that 20 ÷ 4 = 5.

Check the claimed solution

x−8=3 ⇒ x=5x-8=3\ \Rightarrow\ x=5

Check the claimed starting amount. If it fails, find the value that works.

Show the answerHide the answer
x=11x=11

Here x is the starting amount. Add 8 to both sides. Five minus eight is −3, not 3.

Try a new position for x

Choose subtraction or division. Ask what x stands for, solve, then put the answer back into the equation. The game may also use negative values.

2-minute practice · keep going after mistakes · opens in a new tab

Try subtraction equations Try division equations