Does a pair of linear equations always have either a unique solution or infinitely many solutions?
Step-by-step solution
Idea: If the left sides of two equations can be made identical but the right sides are then different, no pair (x, y) can satisfy both.
- Take x − y = 10 … (1) and 10x − 10y = 101 … (2).½ mark
- (1) × 10: 10x − 10y = 100 … (3). Now (2) and (3) have the same left side but different right sides.½ mark
- The same expression cannot equal both 100 and 101, so no (x, y) satisfies both: the pair has no solution.½ mark
- So the answer is no: a pair of linear equations has a unique solution, infinitely many solutions, or no solution.½ mark
Check: Graphically, (1) is y = x − 10 and (2) is y = x − 10.1: same slope, different intercepts, so the lines are parallel and never meet ✓.
Answer to write in the exam
x − y = 10 … (1); 10x − 10y = 101 … (2)
(1) × 10: 10x − 10y = 100 … (3)
(2) − (3): 0 = 1, impossible
∴ No solution; so not always unique or infinitely many
Common mistakes that cost marks
- Concluding “no solution” whenever a calculation looks messy. It is only when you reach a false statement like 0 = 1.
- Thinking (1) and (2) are the same because 101 is close to 100.
How this can come in the exam
Assertion (A): The pair 3x − y = 2, 9x − 3y = 4 has no solution.
Reason (R): For the pair a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, a1a2 = b1b2 ≠ c1c2 means there is no solution.
- Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
Show answer
(A) Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
Write the pair as 3x − y − 2 = 0, 9x − 3y − 4 = 0. Then 39 = 13, −1−3 = 13, −2−4 = 12. So a1a2 = b1b2 ≠ c1c2: the lines are parallel and A is true. R is the rule that gives A, so both are true and R explains A.
Try one yourself
Show that 2x + y = 4 and 6x + 3y = 5 have no solution.
Show answer
3 × the first: 6x + 3y = 12, but the second says 6x + 3y = 5; 12 ≠ 5, so no solution.
More questions like this
- 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.
- Consider the following pairs of linear equations in two variables.
Pair 1: 2x + y = 6 and x − y = 2
Pair 2: x + y = 4 and 2x + 2y = 8
Pair 3: 3x − 2y = 6 and 6x − 4y = 12
Pair 4: x − 2y = 4 and 2x − 4y = 6
For each pair, prepare a table of values (with at least two ordered pairs). Plot the points on the Cartesian plane. Draw the straight lines. - How do we find solutions from the graph? What can we say about the number of solutions when the two lines (i) intersect at a point, (ii) are parallel, (iii) are coincident?
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