Teach Me Maths

Forming and Solving Equations

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How to turn a worded problem, a perimeter, angles or people's ages into an equation, solve it, and use the answer to reply to the question that was asked.

Forming equations in the exam

Forming an equation means writing the equation yourself, from the facts in a question. You choose a letter for the number you don't know, write the facts in algebra, then solve the equation.

In the exam it comes up as a "think of a number" puzzle, a shape whose sides are written using x, angles written using x, or people's ages or amounts of money. These questions are usually worth 2 to 5 marks.

The question may not say "equation" at all. It may say "Work out the value of x", or ask for a length, an angle or an age. There is usually a mark just for writing a correct equation, even if you go wrong later.

Solve the equation with algebra rather than by trying numbers. A wrong guess gets no marks, but a correct equation does.

From words to an equation

  • Choose a letter for the amount you don't know, and say what it stands for.
  • Write each amount in the question as an expression using that letter.
  • Make an equation: put an equals sign between your expression and the number it must equal.
  • Solve the equation, then answer the question that was asked, with units.

For example, Jo thinks of a number, n, and adds 4. Her answer is 11. The equation is n+4=11, so n=7

The order of the operations matters. "Multiply by 3, then add 6" gives 3n+6. "Add 6, then multiply by 3" gives 3(n+6), because the whole of n+6 is multiplied by 3.

Worked example 1: Think of a number

Kofi thinks of a number. He multiplies his number by 5 and then subtracts 8. His answer is 27. What number did Kofi think of?

  1. Let n be Kofi's number.

  2. Your turn. Multiply n by 5, then subtract 8, and make it equal to his answer. Write the equation.

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

    5n-8=27

  3. Add 8 to both sides: 5n=35
  4. Divide both sides by 5: n=7
  5. Check: 7\times5-8=35-8=27
  6. Answer: Kofi thought of 7

Writing amounts in algebra

Key words tell you which operation to use.

  • "5 more than x" or "5 years older" means add: x+5
  • "5 less than x", "5 fewer" or "5 years younger" means subtract: x-5
  • "twice x" or "3 times as many" means multiply: 2x or 3x
  • "half of x" means divide by 2: \frac{x}{2}

Read carefully who is older. If Tariq is x years old and Tariq is 4 years older than Nia, then Nia is younger, so Nia is x-4, not x+4

Write 2x, not x2: the number goes in front of the letter. To find a total, add the expressions and collect like terms. For example, x, x+4 and 2x add up to 4x+4

Worked example 2: Three ages

Amir is x years old. His brother Sami is 3 years younger than Amir. Their father is 4 times as old as Amir. The total of their three ages is 69 years. How old is Sami?

  1. Your turn. Write an expression, in terms of x, for Sami's age.

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

    Sami is younger, so Sami is x-3 years old.

  2. Their father is 4 times as old as Amir, so he is 4x years old.
  3. Add the three ages and make the total 69: x+x-3+4x=69
  4. Collect like terms: 6x-3=69
  5. Add 3 to both sides, then divide by 6: 6x=72, so x=12
  6. Answer: Sami is 12-3=9 years old. Check: 12+9+4\times12=69

Equations from perimeters

The perimeter is the total distance around the edge of a shape. Add the lengths of all the sides, then make the total equal to the perimeter you are given.

A rectangle has two lengths and two widths, so its perimeter is

P=2l+2w

where P is the perimeter, l is the length and w is the width.

When a side is (2x+3) cm, two of those sides make 2(2x+3)=4x+6 cm. Every term in the bracket is doubled.

Don't make one side equal to the whole perimeter, and don't multiply the sides together. Multiplying gives an area, not a perimeter.

Worked example 3: A rectangle's perimeter

A rectangle has length (3x+2) cm and width (x-1) cm. The perimeter of the rectangle is 42 cm. Work out the length of the rectangle.

  1. The perimeter is two lengths plus two widths: 2(3x+2)+2(x-1)
  2. Expand the brackets: 6x+4+2x-2

  3. Your turn. Collect like terms, then write the equation that says the perimeter is 42 cm.

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

    8x+2=42

  4. Subtract 2 from both sides, then divide by 8: 8x=40, so x=5
  5. The length is 3\times5+2=17 cm, and the width is 5-1=4 cm. Check: 2\times17+2\times4=42
  6. Answer: the length is 17 cm.

Equations from angles

When angles are written using x, an angle fact gives you the total to use.

  • Angles on a straight line add up to 180^\circ
  • Angles in a triangle add up to 180^\circ
  • Angles at a point add up to 360^\circ
  • Angles in a quadrilateral add up to 360^\circ

For example, angles of 2x^\circ and (x+30)^\circ on a straight line give 2x+x+30=180. So 3x+30=180, then 3x=150 and x=50

Finding x is often not the end. If the question asks for an angle, put your value of x into that angle's expression. Then check that all the angles add up to the right total.

Worked example 4: Angles in a triangle

In triangle ABC, angle BAC is x^\circ, angle ABC is (2x+15)^\circ and angle ACB is (3x-3)^\circ. Work out the size of angle ABC.

  1. Angles in a triangle add up to 180^\circ: x+2x+15+3x-3=180
  2. Collect like terms: 6x+12=180
  3. Subtract 12 from both sides, then divide by 6: 6x=168, so x=28

  4. Your turn. The question asks for angle ABC, which is (2x+15)^\circ. Work out its size.

    Angle ABC =2\times28+15=71^\circ

  5. Check: the other angles are 28^\circ and 3\times28-3=81^\circ, and 28+71+81=180
  6. Answer: angle ABC is 71^\circ

Problems with more than one step

Harder questions make you use x again after you find it, to work out an age, a length, a cost or a mass. Write down what x stands for at the start, so you know what your answer means.

In an age problem set in the future, everyone gets older by the same number of years. In 5 years, someone aged x will be x+5, and someone aged 2x will be 2x+5, not 2(x+5)

Some questions need two facts. For example, use angles on a straight line to find x, then use angles in a triangle to find the angle that is asked for.

Worked example 5: Ages in the future

Mia is three times as old as her son Leo. In 10 years' time, the total of their ages will be 76 years. How old is Mia now?

  1. Let Leo be x years old now, so Mia is 3x years old now. In 10 years, Leo will be x+10

  2. Your turn. Write an expression, in terms of x, for Mia's age in 10 years' time.

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

    Mia will be 3x+10 years old.

  3. Their total will be 76: x+10+3x+10=76, so 4x+20=76
  4. Subtract 20 from both sides, then divide by 4: 4x=56, so x=14
  5. Leo is 14 and Mia is 3\times14=42. Check: in 10 years they will be 24 and 52, and 24+52=76
  6. Answer: Mia is 42 years old now.

Worked example 6: Two angle facts

A, B and C are points on a straight line, and BCD is a triangle. Angle ABD is (4x+5)^\circ, angle DBC is (x+15)^\circ and angle BCD is 2x^\circ. Work out the size of angle BDC.

  1. ABC is a straight line, so angles ABD and DBC add up to 180^\circ: 4x+5+x+15=180
  2. Collect like terms: 5x+20=180, so 5x=160 and x=32
  3. Angle DBC =32+15=47^\circ. Check: angle ABD =4\times32+5=133^\circ, and 133+47=180
  4. Angle BCD =2\times32=64^\circ

  5. Your turn. Angles in triangle BCD add up to 180^\circ. Work out the size of angle BDC.

    Angle BDC =180-47-64=69^\circ

  6. Answer: angle BDC is 69^\circ

Common mistakes

  • Getting older and younger the wrong way round. If Tom is x and is 5 years older than Amy, then Amy is x-5
  • Leaving an amount out of the total. Add the expression for every person or every side, including the one that is just x
  • Multiplying the expressions together. A total or a perimeter means add the expressions, not multiply them.
  • Making one side equal to the perimeter. The perimeter is all the sides added together.
  • Writing an expression but no equation. Put an equals sign and the total you are given, then solve.
  • Stopping at the value of x. If the question asks for an age, a length or an angle, put x back in to work it out.

Practice questions

More on this topic: Forming and Solving Equations worksheet with full solutions