To learn Simultaneous Equations quickly and correctly, understand why you need two equations: two unknown quantities can be linked by two different conditions, and the solution must satisfy both at the same time. Once you can identify the variables, line up the equations, choose elimination or substitution and check both original statements, the method becomes clear. These eduKateSG Mathematics Quick Notes teach the working step by step with easy examples and a short practice routine.
- Define the unknowns: decide what x and y represent.
- Read both conditions: each equation tells part of the story.
- Choose elimination: remove one variable by adding or subtracting equivalent equations.
- Choose substitution: replace a variable using an equal expression from the other equation.
- Solve one unknown, then the other: state both answers.
- Check twice: substitute the pair into both original equations.
First, learn what a solution pair means. Suppose x + y = 7. Many pairs work: (1, 6), (2, 5) and (4, 3) are examples. That one equation does not yet tell you exactly which pair is intended. Add a second condition, x − y = 1. Now test (4, 3): 4 + 3 = 7 and 4 − 3 = 1. Both conditions are true, so x = 4, y = 3 is a simultaneous solution. A pair that satisfies only one equation is not enough.
Second, use elimination when the equations already have matching or easily matched coefficients. Take x + y = 7 and x − y = 1. Add the two equations: x + x = 2x, while +y and −y cancel. Therefore 2x = 8, so x = 4. Put x = 4 into x + y = 7 to get y = 3. This is not a magic trick: adding two true equalities produces another true equality, and opposite variable terms cancel. Show which equations you added so your method is easy to follow.
Third, use substitution when one equation already gives a variable in terms of the other. Suppose y = 2x − 1 and x + y = 11. Replace y in the second equation: x + (2x − 1) = 11. Simplify: 3x − 1 = 11, then 3x = 12, so x = 4. Substitute into y = 2x − 1: y = 8 − 1 = 7. Check the second equation: 4 + 7 = 11. This is the same principle of equality you already used in ordinary algebra.
Worked example: elimination with different coefficients
Solve 2x + 3y = 18 and 3x + 2y = 17. Multiply the first equation by 3 to get 6x + 9y = 54. Multiply the second equation by 2 to get 6x + 4y = 34. Subtract the second new equation from the first: 5y = 20, so y = 4. Substitute y = 4 into 3x + 2y = 17: 3x + 8 = 17, so x = 3. Check: 2(3) + 3(4) = 18, and 3(3) + 2(4) = 17.
Why multiplying an entire equation is allowed
In elimination, you may multiply both sides of an equation by the same non-zero number without changing its solutions. For example, multiplying 2x + y = 9 by 3 gives 6x + 3y = 27. You must multiply every term on both sides: writing 6x + y = 27 would be wrong. Choose multipliers that make the coefficient of one variable match, then subtract or add the equations as appropriate.
Worked example: turn a short story into two equations
Two adult tickets and three child tickets cost $44. Three adult tickets and two child tickets cost $46. Let a be the adult price and c the child price. Then 2a + 3c = 44 and 3a + 2c = 46. Multiply the first equation by 3: 6a + 9c = 132. Multiply the second by 2: 6a + 4c = 92. Subtract: 5c = 40, so c = $8. Then 3a + 2(8) = 46, giving a = $10. Check the original story: 2(10) + 3(8) = 44, and 3(10) + 2(8) = 46.
Which method should you choose?
- One equation already says x = … or y = …? Substitution often uses fewer steps.
- Matching coefficients with opposite signs? Add equations to eliminate that variable.
- Matching coefficients with the same sign? Subtract equations to eliminate that variable.
- Coefficients do not match yet? Multiply one or both whole equations to create a match.
- Both methods look long? Choose the one with fewer fractions and simpler arithmetic.
- Not sure your pair works? Substitute it into both originals before finishing.
How simultaneous equations appear on a graph
Each linear equation in two variables can be drawn as a straight line. A solution pair is a point where both lines meet. If the lines cross once, they have one solution. If two different lines are parallel, they have no common point and no simultaneous solution. If the equations describe the same line, they have infinitely many common points. The algebraic method finds the same relationship without needing a carefully drawn graph.
What if the equations have no unique solution?
Consider x + y = 5 and 2x + 2y = 10. The second equation says exactly twice the first, so every solution of the first satisfies the second. There are infinitely many solutions. But x + y = 5 and 2x + 2y = 11 contradict each other: doubling the first gives 2x + 2y = 10, not 11. There is no solution. If both variables disappear, read the remaining statement before trying to force a number.
Common mistakes to repair
- Solving only one equation: check that both conditions hold.
- Changing just one term while multiplying an equation: distribute the multiplier across the whole equation.
- Subtracting with incorrect signs: put brackets around an equation being subtracted.
- Forgetting which value belongs to x and y: label the pair clearly.
- Substituting into a line already changed incorrectly: verify using the two original equations.
- Assuming there must always be one solution: watch for identical or contradictory equations.
Mini-test: try all three without the worked example
- Solve 2x + y = 9 and x + y = 5.
- Solve 3a − b = 5 and a + b = 7.
- Solve x + y = 6 and y = 2x.
Answers, with the key step
1. Subtract the second equation from the first to get x = 4; then y = 1. Answer: x = 4, y = 1. 2. Add the equations: 4a = 12, so a = 3 and b = 4. 3. Substitute y = 2x into x + y = 6 to get 3x = 6, so x = 2 and y = 4. In each case, put the pair into both equations to check.
An effective twelve-minute revision routine
- Minutes 1–2: name the two unknowns and explain why one equation may not be enough.
- Minutes 3–5: solve one pair using elimination and show every whole-equation operation.
- Minutes 6–8: solve another pair using substitution.
- Minutes 9–10: check both solution pairs against both original equations.
- Minutes 11–12: write one mistake you could make when changing signs or multiplying an equation, then explain the correction.
The central idea is simple: a simultaneous solution must make two statements true at the same time. Once you understand that, elimination and substitution are just reliable ways to find it.
What to learn next
Read the connected eduKateSG Quick Notes: Linear Equations · Algebra · Coordinate Graphs · Mathematics Word Problems. Study one idea, solve one new example without looking and return tomorrow to check what you remember.
For the broader learning approach, see the immutable reference Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original guide has not been changed.