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20.4 Solving Simultaneous Equations Algebraically

GCSE Maths Foundation · Edexcel · Chapter 20: More Algebra

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: elimination and substitution for two linear equations, and forming equations from problems. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Solve simultaneous equations by adding or subtracting them (elimination)
  • Multiply one or both equations first so that a variable can be eliminated
  • Solve simultaneous equations by substitution
  • Write and solve a pair of simultaneous equations from a word problem

Key words

  • Eliminate: Get rid of one of the letters by adding or subtracting the equations.
  • Coefficient: The number in front of a letter. In 3x + 2y, the coefficient of y is 2.
  • Substitute: Replace a letter with a number or an expression.
  • Solution: The values of x and y that make both equations true.

Adding or subtracting the equations

Line up the equations. If the coefficients of one letter are the same, subtract. If they are the same number with different signs, add. Remember: Same Signs Subtract.

Worked example: x + y = 7 and x − y = 1

The y terms have different signs, so add: 2x = 8, so x = 4
Substitute into x + y = 7: 4 + y = 7, so y = 3
Check in the other equation: 4 − 3 = 1 ✓

Answer: x = 4, y = 3

Worked example: 3x + 2y = 16 and x + 2y = 8

The y terms are both +2y, so subtract: 2x = 8, so x = 4
4 + 2y = 8, so 2y = 4 and y = 2. Check: 12 + 4 = 16 ✓

Answer: x = 4, y = 2

Multiplying first

If no coefficients match, multiply one equation (or both) so that one letter has the same coefficient in each.

Worked example: 2x + y = 11 and x + 3y = 13

Multiply the first equation by 3: 6x + 3y = 33
Subtract x + 3y = 13: 5x = 20, so x = 4
2 × 4 + y = 11, so y = 3. Check: 4 + 9 = 13 ✓

Answer: x = 4, y = 3

Worked example: multiplying both: 3x + 2y = 4 and 5x + 3y = 7

× 3: 9x + 6y = 12. × 2: 10x + 6y = 14
Subtract: x = 2. Then 6 + 2y = 4, so y = −1

Answer: x = 2, y = −1

Substitution

If one equation already says y = … (or x = …), substitute that expression into the other equation.

Worked example: y = 2x + 1 and 3x + y = 16

Replace y: 3x + (2x + 1) = 16
5x + 1 = 16, so 5x = 15 and x = 3
y = 2 × 3 + 1 = 7

Answer: x = 3, y = 7

Word problems

Choose a letter for each unknown, write two equations from the information, solve them, and answer in words with units.

Worked example: cinema tickets

2 adult tickets and 3 child tickets cost £36. 1 adult and 2 child tickets cost £21.
Let an adult ticket cost £a and a child ticket £c: 2a + 3c = 36 and a + 2c = 21
Double the second: 2a + 4c = 42. Subtract the first: c = 6
a + 12 = 21, so a = 9

Answer: Adult £9, child £6

Key fact: Always check your answer in both original equations, and in the words of the problem.

Common mistakes

  • Adding when the signs are the same (or subtracting when they are different).
  • Multiplying only one term of an equation instead of every term, including the number on the right.
  • Mistakes with negatives when subtracting, e.g. 3y − (−2y) = 5y.
  • Finding x and stopping. You must find y as well.

Revision questions

Try these, then check your answers below each one.

Question 1. To solve 2x + 3y = 12 and 5x - 3y = 9, what is the best first step? (Choose from: Add the equations, Subtract the equations, Multiply the first equation by 3, Divide the second equation by 5.)

Answer: Add the equations

Working: +3y and -3y cancel when added: 7x = 21.

Question 2. Is x = 2, y = -1 the solution of 3x + y = 5 and x - y = 3?

Answer: Yes

Working: 6 - 1 = 5 ✓ and 2 - (-1) = 3 ✓.

Question 3. 3 pens and 2 rulers cost £4.10. 1 pen and 2 rulers cost £2.30. How much does one pen cost, in £?

Answer: 0.9

Working: 2p = 1.80, so p = 0.90. A pen costs £0.90.

Related lessons

Continue in the full course

This lesson is section 20.4 (EGCMF2004) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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