Mathématiques
How to Solve Simultaneous Equations
Scale the equations until one letter's coefficients match, add or subtract to remove it, then substitute back. Full worked solution of 3x + 2y = 19 and 4x − 5y = 10.
La réponse en bref
Simultaneous equations are two equations that share the same pair of unknowns, and solving them means finding the one pair of values that satisfies both. Eliminate one letter by scaling the equations until its coefficients match, add or subtract to remove it, then substitute the result back to find the other.
La méthode, étape par étape
Line both equations up in the same order
(1) 3x + 2y = 19 (2) 4x - 5y = 10Putting the x term, the y term and the constant in the same positions in both lines is what makes the next step a column operation rather than a hunt. Numbering the equations matters too — you will refer back to them, and "substitute into (1)" is clearer than pointing at a line.
Choose a letter to eliminate and make its coefficients match
(1) × 5: 15x + 10y = 95 (2) × 2: 8x - 10y = 20We multiply each whole equation by the other's coefficient so the y terms become +10y and −10y. Multiplying an equation through by a number leaves it saying the same thing, which is why this is allowed. Choosing y here rather than x is not arbitrary: the signs are already opposite, so the next step is an addition.
Add or subtract to remove that letter
15x + 10y + 8x - 10y = 95 + 20 → 23x = 115 → x = 5Opposite signs are added; matching signs are subtracted. The purpose is a single equation in one unknown, which is a problem already solved. Adding when the signs match — the most common slip here — leaves both letters in place and the working goes nowhere.
Substitute back into the simpler original equation
3(5) + 2y = 19 → 15 + 2y = 19 → 2y = 4 → y = 2Going back to an original equation rather than one of the scaled versions keeps the numbers small and avoids inheriting an error from the scaling. Either original works; picking the one with the smaller coefficients is a free saving.
Check in the equation you did not use
4(5) - 5(2) = 20 - 10 = 10 ✓Substituting into the equation used for the substitution only proves the arithmetic of that one line. Using the other equation tests both values against information they have not yet been checked against, which is what makes it a genuine check.
What a pair of simultaneous equations is asking
One equation with two unknowns has infinitely many solutions — for 3x + 2y = 19, any x has a y that works. Two equations together usually narrow that to one pair, and that pair is the answer. Geometrically, each equation is a straight line and the solution is the point where they cross.
That picture explains the awkward cases. Two parallel lines never cross, so the equations have no solution and eliminating one letter also removes the other, leaving something false like 0 = 7. Two identical lines cross everywhere, and the same elimination leaves 0 = 0.
The substitution method, and when to use it
Elimination is the default because it handles most pairs with the least writing. Substitution is faster in one specific situation: when one equation already gives a letter on its own, or can be rearranged to do so in one move.
Take y = 2x − 1 and 3x + y = 9. Replacing y in the second equation gives 3x + (2x − 1) = 9, so 5x − 1 = 9, 5x = 10 and x = 2. Then y = 2(2) − 1 = 3. Checking in the second equation: 3(2) + 3 = 9. The pair is x = 2, y = 3.
- Use elimination when both equations are in the form ax + by = c
- Use substitution when one equation already reads y = … or x = …
- Either method gives the same answer — the choice is about arithmetic, not correctness
Turning a word problem into two equations
Most exam marks in this topic are for forming the equations rather than solving them. The pattern is consistent: two unknown quantities, two separate pieces of information about them, one equation from each piece.
Define the letters in writing before anything else — "let c be the cost of one chair in rupees" — because an unlabelled x is where the reasoning marks are lost. Then read each sentence of the question as a separate equation, and only start solving once both are written down.
How we teach this
We insist on the check in the unused equation. Students who check in the equation they substituted into believe wrong answers, and simultaneous equations are a topic where a single sign error produces a plausible-looking pair.
For GCSE students we spend more time on forming the equations than on solving them, because that is where the paper puts the marks. Solving is a procedure; translating a paragraph into two lines of algebra is the skill being examined.
Questions fréquentes
When do you add and when do you subtract the equations?
Look at the signs of the terms you are eliminating. If they are opposite, such as +10y and −10y, add the equations. If they match, such as +10y and +10y, subtract. Getting this the wrong way round leaves both letters in place, which is the signal to check the signs.
What is the difference between elimination and substitution?
Elimination scales both equations so one letter cancels when you add or subtract them. Substitution rearranges one equation to give a letter on its own and puts that expression into the other. Both give the same answer; elimination is usually less writing unless one equation already reads y = something.
Why does my working end with 0 = 5?
Because the two equations describe parallel lines and there is no pair of values that satisfies both. It is not an error — it is the answer, and the mark is for saying there is no solution. If you end with 0 = 0 instead, the two equations are the same line and there are infinitely many solutions.
Which original equation should I substitute back into?
Whichever has the smaller coefficients, and always an original rather than one you scaled. Using a scaled equation carries any multiplication error forward into the second answer, and it is harder to spot there than in the first.
Can simultaneous equations have three unknowns?
Yes, but then three equations are needed — as a rule, you need as many independent equations as unknowns. The method is the same: eliminate one letter to reduce three equations to two, then solve the pair. That extends beyond GCSE, where two unknowns is the standard case.
Sources
- 5.3 Solve Systems of Equations by Elimination — Elementary Algebra 2e — OpenStax, Rice University
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