Simultaneous equations are two equations with two unknowns that must both be true at the same time. To solve by elimination, make the coefficients of one unknown match, then add the equations if the signs are different or subtract if they are the same, so that unknown disappears. Solve for the other unknown, then substitute back to find the first.
Elimination method
- Line up the equations with x terms, y terms and numbers in columns.
- If needed, multiply one or both equations so that one unknown has matching coefficients.
- Different signs: add. Same signs: subtract.
- Solve for the remaining unknown, then substitute to find the other.
- Check both values in the equation you did not use for substituting.
What does the solution mean?
The solution is the pair of values that makes both equations true. Each linear equation is a straight line on a graph, and the solution is the point where the two lines cross. That is why simultaneous equations can also be solved graphically, by reading the intersection point.
Adding or subtracting
Look at the coefficients of one unknown. If they are equal and have opposite signs, such as +y and −y, adding the equations removes that unknown. If they are equal with the same sign, such as 2y and 2y, subtracting removes it. A simple way to remember: Same Signs Subtract.
Multiplying first
If no coefficients match, multiply one equation, or both, by a number so that they do. For 2x + 3y = 23 and 5x − y = 15, multiplying the second equation by 3 gives 15x − 3y = 45, and adding then removes y. Multiply every term in the equation, including the number on the right.
Substitution method
When one equation already gives one unknown in terms of the other, such as y = 2x − 1, substitute it into the other equation. This method is essential at Higher tier, where one equation may be quadratic.
Worded problems
Many questions describe two purchases or two totals. Choose letters for the unknown prices, write one equation for each piece of information, then solve. Give the answer in context with units, such as “an adult ticket costs £8”.
Worked examples
Example 1. Solve 2x + y = 7 and x − y = 5.
- The y terms have opposite signs, so add: 3x = 12, so x = 4.
- Substitute into x − y = 5: 4 − y = 5, so y = −1.
- Check in 2x + y = 7: 8 − 1 = 7.
Answer: x = 4, y = −1.
Example 2. Solve 3x + 2y = 22 and x + 2y = 10.
- The y terms are the same, so subtract: 2x = 12, so x = 6.
- Substitute: 6 + 2y = 10, so y = 2.
Answer: x = 6, y = 2.
Example 3. Solve 2x + 3y = 23 and 5x − y = 15.
- Multiply the second equation by 3: 15x − 3y = 45.
- Add to the first: 17x = 68, so x = 4.
- Substitute: 5(4) − y = 15, so y = 5.
Answer: x = 4, y = 5.
Example 4. Solve 3x + 4y = 29 and 2x + 3y = 20.
- Multiply the first by 2 and the second by 3: 6x + 8y = 58 and 6x + 9y = 60.
- Subtract: y = 2.
- Substitute: 3x + 8 = 29, so x = 7.
Answer: x = 7, y = 2.
Example 5. Three adult tickets and two child tickets cost £36. One adult ticket and two child tickets cost £20. Find the cost of each ticket.
- 3a + 2c = 36 and a + 2c = 20.
- Subtract: 2a = 16, so a = 8.
- Substitute: 8 + 2c = 20, so c = 6.
Answer: Adult £8, child £6.
Common mistakes
- Adding when you should subtract. Same signs subtract; different signs add.
- Multiplying only part of an equation. Multiply every term, including the number on the right.
- Sign errors when subtracting negatives. Subtracting −y is the same as adding y.
- Stopping after one unknown. Substitute back to find the second value.
- Not checking. Substitute both values into the other equation to confirm.
How simultaneous equations are examined
Linear simultaneous equations are tested on both tiers of Edexcel GCSE (1MA1) and the International GCSE (4MA1); Higher tier adds pairs where one equation is quadratic. Questions ask you to:
- solve a pair of linear equations, showing clear working
- form and solve equations from a context such as prices
- solve graphically by reading the intersection of two lines
- solve a linear and a quadratic equation together (Higher)
Pairs of equations to solve, with worked answers, are on the simultaneous equations worksheet. The algebra engine can generate more pairs whenever you need them.
This uses linear equations, and the graphical method links to linear graphs.
Frequently asked questions
What are simultaneous equations?
Two or more equations that share the same unknowns and must all be true at once.
When do you add and when do you subtract?
Add if the matching coefficients have different signs; subtract if they have the same sign.
What does the solution mean on a graph?
It is the point where the two lines intersect.