Does a pair of linear equations always have a unique solution? Can you analyse this using the Elimination Method? While subtracting the equations to eliminate a particular variable, if the other variable remains, then clearly there is a unique solution. But what if the second variable gets eliminated too while eliminating the first variable? Can you think of a situation where this happens?
Step-by-step solution
Idea: Elimination subtracts matched equations. Both variables vanish together exactly when the coefficients of x and y are in the same ratio; what is left is a statement about the constants only.
- Both variables disappear together when, after matching the x-coefficients, the y-coefficients also match, i.e. when a1a2 = b1b2.1 mark
- Case 1: x − y = 10 and 10x − 10y = 100. Multiply the first by 10: 10x − 10y = 100. Subtract: 0 = 0. Always true, so every solution of x − y = 10 works: infinitely many solutions.1 mark
- Case 2: x − y = 10 and 10x − 10y = 101. Multiply the first by 10 and subtract: 0 = 1. Impossible, so no solution.½ mark
- So a pair does not always have a unique solution: it can have one, infinitely many, or none.½ mark
Check: In case 1, (12, 2) and (0, −10) both satisfy both equations ✓. In case 2, 10x − 10y cannot be both 100 and 101 ✓.
Answer to write in the exam
Both variables vanish when a1a2 = b1b2
e.g. x − y = 10, 10x − 10y = 100: 10 × (1) − (2) ⇒ 0 = 0 ⇒ infinitely many solutions
e.g. x − y = 10, 10x − 10y = 101: ⇒ 0 = 1 ⇒ no solution
∴ A pair need not have a unique solution
Common mistakes that cost marks
- Reading 0 = 0 as “the solution is x = 0, y = 0″. It means the equations are the same, with infinitely many solutions.
- Thinking 0 = 1 means a calculation error. If the working is right, it shows there is no solution.
How this can come in the exam
Eliminating x from 2x + 3y = 5 and 4x + 6y = 9 leads to
- a unique solution
- 0 = 0
- 0 = 1
- y = 1
Show answer
(C) 0 = 1
2 × (1): 4x + 6y = 10; subtract (2): 0 = 1, so no solution.
Try one yourself
What happens when you eliminate x from 3x − y = 2 and 6x − 2y = 4?
Show answer
2 × (1) − (2) gives 0 = 0: infinitely many solutions (same line).
More questions like this
- How many solutions are there to the equations 10x − 5y = 20 and 4x − 2y = 8?
Give 3 more examples of pairs of equations that lead to 0 = 0 after subtraction, and therefore have infinitely many solutions. Can you find a simple rule to check when this happens? - What if one of a2, b2, c2 is zero?
- Does a pair of linear equations always have either a unique solution or infinitely many solutions?
- Give 3 more examples of pairs of equations that have no solution. Can you find a simple rule to check when this will happen?
- Are the converses of the above statements true?
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