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One-step and two-step linear equations

Using the balancing method to find an unknown.

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In Lesson 5.3 you knew the number, so you could work out the answer. Now it is the other way round: you know the answer, and you find the number the letter stands for. In this lesson you will learn how to:

  • say what an equation is
  • use the balancing method: do the same to both sides
  • solve one-step equations by undoing
  • solve two-step equations by undoing in reverse order
  • check an answer by putting it back in
  • write an equation from words, then solve it
Dark blue: numbers Red: letters, the numbers we are looking for

1. What is an equation?

Key idea An equation has an = sign. It says that both sides are worth the same, like a balance scale that is level. In n + 3 = 8, only one number works: n = 5. Finding that number is called solving the equation.

A balance scale: n + 3 = 8

n

A bag holds n sweets. The bag and 3 more sweets weigh the same as 8 sweets, so n + 3 = 8. Which number makes it balance? 5 + 3 = 8, so n = 5. Try 4 or 6 and the scale tips.

Expression or equation?

n + 3 is an expression (Lesson 5.3). There is no = sign, so n could be any number.

n + 3 = 8 is an equation. n has to be 5.

You have seen this before

A missing number puzzle like   + 3 = 8 is an equation too. In algebra, a letter takes the place of the box.

Watch out The = sign means “is worth the same as”. It does not mean “the answer comes next”. 8 = n + 3 is the same equation as n + 3 = 8, written the other way round.

2. The balancing method

Key idea To solve an equation, get the letter on its own. Whatever you do to one side, do the same to the other side, and it stays balanced. Use the opposite to undo: take away undoes add, add undoes take away, divide undoes multiply, and multiply undoes divide.

Example: solve n + 3 = 8

  1. n has 3 added. Take away 3 from both sides:n = 8 − 3
  2. So:n = 5
  3. Check: put 5 back in.5 + 3 = 8 ✓
Answern = 5

The four kinds of one-step equation

EquationUndo itAnswerCheck
n + 7 = 15Take away 7 from both sidesn = 88 + 7 = 15
n − 4 = 9Add 4 to both sidesn = 1313 − 4 = 9
3n = 21Divide both sides by 3n = 73 × 7 = 21
n ÷ 4 = 6 or n4 = 6Multiply both sides by 4n = 2424 ÷ 4 = 6

Sharing both sides: 3n = 12

nnn

Three bags balance 12 sweets. Share both sides into 3 equal groups. One bag balances 4 sweets, so n = 4. Sharing into 3 groups is the same as dividing both sides by 3.

Watch out Do the opposite, not the same. For n + 7 = 15, adding 7 gives 22, which is wrong. Check: 22 + 7 is 29, not 15. Putting your answer back in always shows a mistake like this.

3. Two-step equations

Key idea In 3n + 4 = 19, two things happened to n: first it was multiplied by 3, then 4 was added. To undo them, work backwards: undo the last step first. It is like getting dressed. Socks go on first, then shoes, but shoes come off first.

The machine, forwards and backwards

Going forwards, n is multiplied by 3, then 4 is added, to make 19. Going backwards from 19, take away 4, then divide by 3. So n = 5.

Example: solve 3n + 4 = 19

  1. Undo the + 4 first. Take away 4 from both sides:3n = 15
  2. Then undo the × 3. Divide both sides by 3:n = 5
  3. Check: put 5 back in.3 × 5 + 4 = 15 + 4 = 19 ✓
Answern = 5

A division: n2 − 3 = 5

Undo the − 3 first, then the ÷ 2. Check: 16 ÷ 2 = 8, and 8 − 3 = 5.

Brackets: 2(n + 1) = 14

The brackets mean the 1 was added first, then it was multiplied by 2. So undo the × 2 first. Check: 6 + 1 = 7, and 2 × 7 = 14.

Watch out In 3n + 4 = 19, do not divide by 3 first. The 4 was never multiplied by 3, so dividing gets muddled. Undo the adding or taking away first, then the multiplying or dividing.

4. Writing an equation from words

Key idea Use a letter for the number you do not know. Write what happens to it as an expression, as in Lesson 5.3, then put = the answer. Now you have an equation to solve.

Example: I think of a number. I double it and add 5. The answer is 23. What is my number?

  1. Call the number n. Double it and add 5:2n + 5
  2. The answer is 23:2n + 5 = 23

Check: double 9 is 18, and 18 + 5 = 23. ✓

AnswerThe number is 9.

A story: the notebooks

Ella buys 4 notebooks and a pen that costs £3. She spends £19. Each notebook costs £c. The equation is 4c + 3 = 19. Take away 3 from both sides: 4c = 16. Divide both sides by 4: c = 4. Each notebook costs £4.

Watch out Answer the question that was asked. If it asks for the cost of a notebook, write “£4”, not just c = 4. And check your answer with the words, not only with your equation: if the equation is wrong, the check with the words will show it.

5. Solve it, then check it

Key idea Write each step on a new line, with what you did to both sides underneath. When n is on its own, put your answer back into the first equation. If both sides match, you are right.

The letter on the right: 19 = 3n + 4

This is the same as 3n + 4 = 19, written the other way round. Solve it in the same way: n = 5.

Taking the letter away: 20 − n = 13

What do you take away from 20 to leave 13? 20 − 13 = 7, so n = 7. Check: 20 − 7 = 13. Adding 13 to 20 would give 33, which does not check.

An answer below zero: n + 9 = 4

Take away 9 from both sides: n = 4 − 9. On the number line from Lesson 1.4, start at 4 and jump 9 down. You land on −5. Check: −5 + 9 = 4.

Watch out Always check in the first equation, the one in the question. A slip in the middle of your working will not show up if you check a line you wrote yourself.

The mistakes that cost marks

  1. Doing the same instead of the opposite. Solving n + 7 = 15 by adding 7 to get 22. undo the + 7 by taking away 7: n = 8.
  2. Changing only one side. Taking 4 from the left of 3n + 4 = 19 but not from the right. do the same to both sides: 3n = 15.
  3. Undoing in the wrong order. Dividing 3n + 4 = 19 by 3 first. undo the + 4 first, then the × 3.
  4. Dividing only part of a side. Turning 4n + 8 = 36 into n + 8 = 9. take away the 8 first: 4n = 28, then n = 7.
  5. Mixing up the brackets. Solving 2(n + 1) = 14 by taking away 1 first. the × 2 happened last, so undo it first: n + 1 = 7, then n = 6.
  6. Adding when the letter is taken away. Solving 20 − n = 13 as n = 33. what do you take from 20 to leave 13? 20 − 13 = 7.
  7. Not checking. Stopping as soon as you have a number. put it back into the first equation. Both sides must match.
  8. Answering with the letter. Writing c = 4 when the question asks for a cost. answer in words, with units: each notebook costs £4.

Quick checklist

Tick each one when you can do it without help.

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