Teach Me Maths

Equations with Brackets and Unknowns on Both Sides

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How to solve equations with the letter on both sides, how to expand a bracket before you start, how to form an equation when two lengths or two angles in a shape are equal, and how to check your answer.

What the exam asks

An equation with unknowns on both sides has the letter on the left and on the right, as in 6x+1=2x+13. Solving it means finding the value of x that makes the two sides equal.

In the exam you are usually asked to solve an equation such as 5x-4=2x+11, for 2 or 3 marks. Some have a bracket to expand first. Harder questions ask you to form the equation yourself, when two lengths or two angles in a shape are equal.

Write each step on a new line. Marks are given for correct steps, such as expanding a bracket or getting the x-terms together, even if the final answer goes wrong.

Keep the equation balanced

An equation is like a balance: both sides weigh the same. If you take the same amount off both sides, it still balances.

Take one bag off each side: 2x+4=10. Then take 4 counters off each side: 2x=6. So each bag holds 3 counters, and x=3

The same steps work for any equation like this:

  • Get the x-terms together on one side, usually by subtracting the smaller x-term from both sides.
  • Move the numbers to the other side, doing the same to both sides.
  • Divide to find x.
  • Check by putting your answer into both sides of the original equation.

Worked example 1: Unknowns on both sides

Solve 6x+1=2x+13

  1. Subtract 2x from both sides: 4x+1=13
  2. Subtract 1 from both sides: 4x=12

  3. Your turn. Divide both sides by 4. What is x?

    x=12\div4=3

  4. Check: 6\times3+1=19 and 2\times3+13=19. Both sides agree.
  5. Answer: x=3

When an x-term is taken away

Sometimes an x-term is being subtracted, as in x+9=15-2x. To get rid of the -2x, add 2x to both sides, because -2x+2x=0. This gives 3x+9=15

Like terms have the same letter, such as x and 2x. Only like terms can be collected: x+2x=3x, but 3x+9 stays as it is. It is not 12x.

Worked example 2: Choosing which side to use

Solve 3x+14=7x-2

  1. The smaller x-term is 3x. Subtract 3x from both sides: 14=4x-2
  2. Add 2 to both sides: 16=4x, which means the same as 4x=16

  3. Your turn. What is x?

    x=16\div4=4

  4. Check: 3\times4+14=26 and 7\times4-2=26
  5. Answer: x=4

Worked example 3: Adding an x-term to both sides

Solve 2x-7=18-3x

  1. The right side has -3x. Add 3x to both sides: 2x+3x-7=18

  2. Your turn. Simplify 2x+3x

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    2x+3x=5x, so 5x-7=18

  3. Add 7 to both sides: 5x=25
  4. Divide both sides by 5: x=5
  5. Check: 2\times5-7=3 and 18-3\times5=3
  6. Answer: x=5

Expand brackets first

A number outside a bracket multiplies every term inside it. So 4(x+3)=4x+12 and 4(x-3)=4x-12

Expand the bracket before you move anything. In 2(x+6)=3x+5 you cannot take 6 off both sides, because the 6 is inside the bracket and is multiplied by 2. Expand first: 2x+12=3x+5

Watch the signs. 3(5-x) is 3\times5 and 3\times(-x), which gives 15-3x

Worked example 4: A bracket on one side

Solve 5(x-2)=2x+8

  1. Your turn. Expand the bracket 5(x-2). The 5 multiplies both terms.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    5(x-2)=5x-10, so the equation is 5x-10=2x+8

  2. Subtract 2x from both sides: 3x-10=8
  3. Add 10 to both sides: 3x=18
  4. Divide both sides by 3: x=6
  5. Check: 5(6-2)=5\times4=20 and 2\times6+8=20
  6. Answer: x=6

Worked example 5: An answer that is not a whole number

Solve 6x-1=2(x+4)

  1. Expand the bracket on the right: 6x-1=2x+8
  2. Subtract 2x from both sides: 4x-1=8
  3. Add 1 to both sides: 4x=9

  4. Your turn. Divide both sides by 4. Give x as a decimal.

    x=9\div4=2.25

  5. Check: 6\times2.25-1=12.5 and 2(2.25+4)=2\times6.25=12.5
  6. Answer: x=2.25. An answer does not have to be a whole number: x=2\frac{1}{4} or x=\frac{9}{4} is also correct.

Forming an equation

Some questions do not give you the equation. Instead they show a shape and tell you that two of its lengths or two of its angles are equal. Each one is written as an expression in x.

Put an equals sign between the two expressions that are equal. Then solve the equation.

Read the question again at the end. If it asks for a length, an angle or a perimeter, put your value of x back in to work it out.

Worked example 6: Two sides that are equal

Triangle PQR is isosceles, with PQ=PR. PQ=3x+4, PR=5x-6 and QR=12. All the lengths are in centimetres. Work out the perimeter of the triangle.

  1. PQ=PR, so 3x+4=5x-6
  2. Subtract 3x from both sides: 4=2x-6
  3. Add 6 to both sides: 10=2x, so x=5

  4. Your turn. Put x=5 into 3x+4. How long is PQ, in centimetres?

    PQ=3\times5+4=19 cm. Check: PR=5\times5-6=19 cm too.

  5. Perimeter =19+19+12=50 cm
  6. Answer: 50 cm

Common mistakes

  • Changing only one side. Whatever you add, subtract or divide by, do it to both sides.
  • Subtracting when you should add. To remove -3x, add 3x to both sides.
  • Taking away a number that is inside a bracket. Expand the bracket first: 2(x+6) is 2x+12.
  • Multiplying only the first term in a bracket. 5(x-2) is 5x-10, not 5x-2.
  • Collecting unlike terms. 3x+9 cannot be written as 12x.
  • Stopping at a line such as 4x=12. Divide to find one x: x=3

Practice questions

More on this topic: Equations with Brackets and Unknowns on Both Sides worksheet with full solutions