Lesson 8.5 · 30 min
Simultaneous Equations
Solving pairs of equations by elimination — adding, subtracting, and multiplying first when the coefficients do not match.
In this lesson you will learn to…
- Explain what the solution of a pair of simultaneous equations means
- Solve by adding or subtracting the equations to eliminate one unknown
- Multiply one or both equations first so that a coefficient matches
- Form and solve simultaneous equations from a word problem
Why this matters
Many real problems have two unknowns tied together by two facts: 5 teas and a coffee cost 7 dollars; 2 teas and a coffee cost 4 dollars. Simultaneous equations let you untangle them. Graphically, the solution is the point where two straight lines cross.
The café problem
Ahmad ordered 3 more teas than Nadia and paid 3 dollars more, so subtracting gives , and a tea costs 1 dollar. Then , so a coffee costs 2 dollars. That is the elimination method: remove one letter so you can solve for the other.
Choosing what to do
| Equations | What to do |
|---|---|
| and | same sign on → subtract |
| and | opposite signs on → add |
| and | multiply the second by 2, then add |
| and | multiply both, then subtract |
Stephan’s mistake. For and he subtracted because the terms are both positive — but that gives , which still has both letters. The terms have opposite signs, so add: .
Key rules and formulas
Two unknowns need two equations
One equation such as has infinitely many solutions. The solution of a pair is the one that works in both.Same sign → subtract
If the coefficients of one letter are equal with the same sign, subtract the equations: minus gives .Opposite signs → add
If they are equal but with opposite signs, add: plus gives .No match? Multiply first
Multiply one (or both) equations so a pair of coefficients matches. Then substitute back to find the other letter, and check in the equation you did not use.
Worked examples
Subtracting
Try it yourself first, then reveal the solution one step at a time.
Multiplying one equation
Try it yourself first, then reveal the solution one step at a time.
Multiplying both equations
Try it yourself first, then reveal the solution one step at a time.
A word problem
Try it yourself first, then reveal the solution one step at a time.
Practice: simultaneous equations
10 questions
Question 1 of 10Fill the gapsFoundation Solve and .Question 2 of 10Multiple choice Which is the solution of and ?Question 3 of 10Match the pairs Match each pair of equations to the best first step.Pick an answer, then choose the row it belongs to. You can also drag it.
- and
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Question 4 of 10Step by step Solve by adding.Solve [1] and [2].- Which letter can you eliminate straight away?
Question 5 of 10Fill the gaps Solve and .Question 6 of 10Fill the gaps Solve and .Question 7 of 10Fill the gaps The sum of two numbers is 19 and their difference is 5. Find the numbers.Larger: Smaller:
Question 8 of 10Step by step Form and solve simultaneous equations.Five small glasses and three large glasses hold 2450 ml. Two small and three large hold 1700 ml.- Subtracting the equations gives
Question 9 of 10Fill the gapsChallenge At a farm, sheep and children have 16 heads and 40 feet altogether. How many of each?Children: Sheep:
Question 10 of 10Fill the gapsChallenge Emily buys 23 bottles of water. Still water costs 50 cents and sparkling costs 75 cents. She spends 13.50 dollars. How many of each did she buy?Still: Sparkling: