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?
- 1. How many solutions are there to the equations 10x − 5y = 20 and 4x − 2y = 8?
- 2. Give 3 more examples of pairs of equations that lead to 0 = 0 after subtraction, and therefore have infinitely many solutions.
- 3. Can you find a simple rule to check when this happens?
Step-by-step solution
Idea: If one equation is a constant multiple of the other (every coefficient and the constant multiplied by the same k ≠ 0), they are the same equation and share all their solutions.
1. How many solutions are there to the equations 10x − 5y = 20 and 4x − 2y = 8?
- 10x − 5y = 20 is 5 × (2x − y = 4) and 4x − 2y = 8 is 2 × (2x − y = 4).½ mark
- Multiply the first by 2 and the second by 5: both become 20x − 10y = 40; subtracting gives 0 = 0. They are the same line, so there are infinitely many solutions, e.g. (2, 0), (0, −4), (3, 2).½ mark
2. Give 3 more examples of pairs of equations that lead to 0 = 0 after subtraction, and therefore have infinitely many solutions.
- Multiply any equation by a non-zero number to get its partner: (a) x + y = 3 and 2x + 2y = 6 (× 2); (b) 3x − y = 1 and 9x − 3y = 3 (× 3); (c) x − 4y = 2 and −2x + 8y = −4 (× −2).1 mark
3. Can you find a simple rule to check when this happens?
- For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0: this happens when a1 = ka2, b1 = kb2, c1 = kc2 for some k ≠ 0, that is, a1a2 = b1b2 = c1c2. For the given pair: 104 = −5−2 = −20−8 = 52 ✓.1 mark
Check: (3, 2) satisfies both given equations: 30 − 10 = 20 ✓ and 12 − 4 = 8 ✓.
Answer to write in the exam
1.
10x − 5y = 20 ⇒ 2x − y = 4 (÷ 5)
4x − 2y = 8 ⇒ 2x − y = 4 (÷ 2)
Same equation ⇒ subtraction gives 0 = 0
∴ Infinitely many solutions
2.
x + y = 3 and 2x + 2y = 6
3x − y = 1 and 9x − 3y = 3
x − 4y = 2 and −2x + 8y = −4
3.
Rule: a1a2 = b1b2 = c1c2 ⇒ infinitely many solutions
Given pair: 104 = −5−2 = −20−8 = 52
Common mistakes that cost marks
- Answering “one solution” after finding a single pair such as (2, 0). Check whether the equations are multiples of each other first.
- Giving an example where only the x– and y-coefficients are multiplied, not the constant (that gives parallel lines, no solution).
How this can come in the exam
Which pair has infinitely many solutions?
- x + 2y = 4, 2x + 4y = 9
- x + 2y = 4, 3x + 6y = 12
- x + 2y = 4, 2x + y = 4
- x + 2y = 4, x − 2y = 4
Show answer
(B) x + 2y = 4, 3x + 6y = 12
13 = 26 = 412.
Try one yourself
For what k do 3x + 5y = 7 and 6x + 10y = k have infinitely many solutions?
Show answer
36 = 510 = 7k ⇒ k = 14.
More questions like this
- 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?
- Can there be methods other than elimination and substitution to reduce a pair of linear equations in two variables to a linear equation in one variable? If so, describe them.
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