Introduction to Simultaneous Equations
Simultaneous equations are two or more equations that share the same unknowns, where a solution is a set of values that makes every equation true at the same time. The elimination method solves two linear equations with two unknowns by adding or subtracting the equations to remove one unknown, then solving for the other and substituting back; graphically, the solution is the point where the two lines cross.
Before you start
Make sure you're comfortable with these topics first:
Method
- Label the two equations (1) and (2) so you can refer back to them clearly.
- Check whether the coefficient of one of the letters is the same (or a simple multiple) in both equations.
- If needed, multiply one or both equations by a number so that the coefficients of one letter are equal (to subtract) or equal and opposite (to add).
- Add or subtract the two equations to eliminate that letter, leaving a single equation in one unknown.
- Solve this equation, then substitute the value back into either original equation to find the other unknown.
- Check both values by substituting them into the equation you did not use for the substitution step.
Worked example
Solve the simultaneous equations 2x + y = 11 and x + y = 7.
- Label the equations: (1) 2x + y = 11, (2) x + y = 7.
- The coefficient of y is 1 in both equations, so subtract equation (2) from equation (1): (2x + y) - (x + y) = 11 - 7, giving x = 4.
- Substitute x = 4 into equation (2): 4 + y = 7, so y = 3.
- Check in equation (1): 2(4) + 3 = 8 + 3 = 11, which matches, so the solution is x = 4, y = 3.
Practice questions
Try each question, then tap to reveal the answer.
Q1Solve x + y = 10 and x - y = 2.Show answer
Answer: Adding: 2x = 12, so x = 6, then y = 4.
Q2Solve 3x + y = 17 and x + y = 7.Show answer
Answer: Subtracting: 2x = 10, so x = 5, then y = 2.
Q3Solve 2x + 3y = 16 and 2x + y = 8.Show answer
Answer: Subtracting: 2y = 8, so y = 4, then 2x = 4 and x = 2.
Q4Solve x + 2y = 13 and x + y = 8.Show answer
Answer: Subtracting: y = 5, then x = 3.
Q5Solve x + y = 9 and x - y = 1.Show answer
Answer: Adding: 2x = 10, so x = 5, then y = 4.
Q6Solve 4x + y = 19 and x + y = 7.Show answer
Answer: Subtracting: 3x = 12, so x = 4, then y = 3.
Q7Solve 2x + y = 13 and x + 3y = 14.Show answer
Answer: Multiply the first equation by 3: 6x + 3y = 39. Subtract the second equation: 5x = 25, so x = 5, then y = 13 - 2(5) = 3.
Q8Solve 3x + 2y = 12 and 3x - y = 3.Show answer
Answer: Subtracting: 3y = 9, so y = 3, then 3x = 6 and x = 2.
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Solve the simultaneous equations 3x + y = 19 and x + y = 9. Show your working.
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A cinema sells adult tickets at a pounds each and child tickets at c pounds each. Two adults and three children pay 37 pounds in total, so 2a + 3c = 37. One adult and three children pay 26 pounds in total, so a + 3c = 26. Find the price of an adult ticket and the price of a child ticket.
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Free printable worksheet
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This topic is chapter 32 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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