Simultaneous equations

Simultaneous equations are two or more equations with the same unknowns, solved together to find values that make all of them true at once. For two linear equations in x and y, there are three ways to solve them: elimination, where you add or subtract the equations to remove one letter; substitution, where you put one equation into the other; and graphs, where the solution is the point where the two lines cross. In the US they are called systems of equations.

What are simultaneous equations?

One equation with two unknowns, such as x + y = 10, has endless solutions: 1 and 9, 4 and 6, 2.5 and 7.5, and so on. A second equation narrows it down. If you also know x − y = 4, only one pair works for both: x = 7 and y = 3. Solving simultaneous equations means finding that pair.

Each linear equation is a straight line on a graph. Every point on a line fits its equation, so the point where two lines cross fits both. That is why most pairs of linear simultaneous equations have exactly one solution.

How to solve simultaneous equations by elimination

Elimination is the method used most at GCSE. The idea is to add or subtract the equations so that one letter disappears, leaving one equation with one unknown.

1. Line up the equations so the x terms, y terms and numbers are in columns. Label them (1) and (2).
2. Match the coefficients of one letter, if they are not already the same size, by multiplying one or both equations.
3. Add or subtract to eliminate that letter. If the signs are different, add. If the signs are the same, subtract. (A handy memory aid: SSS, Same Sign Subtract.)
4. Solve the equation that is left.
5. Substitute that value back into one of the original equations to find the other letter.
6. Check both values in the other original equation.

Example 1: add to eliminate

Solve 3x + 2y = 17 and 5x − 2y = 7.

The y terms are +2y and −2y: the same size with different signs, so add the equations.

So x = 3. Substitute into the first equation: 9 + 2y = 17, so 2y = 8 and y = 4.

x = 3, y = 4. Check in the second equation: 15 − 8 = 7. Correct.

Example 2: subtract to eliminate

Solve 4x + 3y = 26 and 4x + y = 14.

Both x terms are +4x, the same sign, so subtract the second equation from the first.

So y = 6. Then 4x + 6 = 14, so 4x = 8 and x = 2, y = 6.

Example 3: multiply first

Solve 5x + 3y = 9 and 3x − 2y = 13.

No coefficients match yet. To eliminate y, make both y coefficients 6: multiply the first equation by 2 and the second by 3.

The signs are different, so add: 19x = 57, which gives x = 3. Substitute into 5x + 3y = 9: 15 + 3y = 9, so 3y = −6 and x = 3, y = −2.

Check in 3x − 2y = 13: 9 + 4 = 13. Correct.

Now try one. Solve the pair, then press Mark to check both values in the course.

Solving simultaneous equations by substitution

Substitution is quickest when one equation already says y = ... or x = .... Replace that letter in the other equation, solve, then substitute back.

Example 4: substitution

Solve y = 2x − 1 and 3x + 2y = 19.

Replace y in the second equation with 2x − 1:

So x = 3, and y = 2 × 3 − 1 = 5. x = 3, y = 5.

Substitution is also the method you need when one equation is not linear, such as x² + y² = 25 with y = x + 1. Those are quadratic simultaneous equations, a GCSE Higher topic, and you substitute the linear equation into the quadratic one.

Solving simultaneous equations graphically

Draw both lines on the same axes. The coordinates of the point where they cross are the solution. For y = x + 1 and y = 7 − 2x, the lines cross at (2, 3), so x = 2 and y = 3.

The graphical method shows you what the solution means, but reading a graph usually only gives an approximate answer. GCSE questions ask for "approximate solutions using a graph" for exactly that reason. Use algebra when you need an exact answer. The Simultaneous Equations Calculator draws both lines and solves the pair, so you can see where they meet.

When there is no solution or infinitely many

Not every pair of equations has one solution.

In elimination, a pair with no solution ends with something impossible such as 0 = 4. A pair with infinitely many ends with 0 = 0.

Which method should you use?

Simultaneous equations word problems

Many exam questions describe a situation in words. Choose letters for the two unknowns, write two equations, then solve as usual.

Example 5: sports equipment

A school pays £94 for 6 footballs and 4 netballs. Another order of 3 footballs and 5 netballs costs £77. Find the price of each ball.

Let f be the price of a football and n the price of a netball.

Doubling the second equation makes the f coefficients match:

Same sign, so subtract the first equation from this one: 6n = 60, which gives n = 10. Then 6f + 40 = 94, so 6f = 54 and f = 9. A football costs £9 and a netball £10. Check: 3 × 9 + 5 × 10 = 27 + 50 = 77.

Common mistakes

• Adding when you should subtract. Same sign, subtract; different signs, add. Write the signs in before you start.
• Multiplying only one side. When you multiply an equation by 3, multiply every term, including the number on the right.
• Sign errors when subtracting a negative. −2y − (−6y) is +4y. Put brackets round the equation you are subtracting.
• Stopping after one letter. You need both values. Substitute back to find the second.
• Skipping the check. Put both values into the equation you did not use for substitution. If it does not work, there is a slip somewhere.

Simultaneous equations at GCSE

In AQA's GCSE Maths specification, solving two linear simultaneous equations algebraically, and finding approximate solutions from a graph, is on both Foundation and Higher papers. Simultaneous equations where one is linear and one is quadratic are Higher tier only. You also need to form your own pair of equations from a problem in words and interpret the answer. There is no formula to learn, so nothing about the method is on the formula sheet.

Practise in a short timed paper

Simultaneous equations come up on most GCSE Maths papers, often as a word problem. GCSE Maths Foundation and GCSE Maths Higher each give you a short timed paper every day, with 8 questions worth 25 to 35 marks. In the US, the same practice opens as Pre-Algebra and Algebra 1. To learn the topic from the start, the Algebra course shows two lines meeting and then builds up the algebra.

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