Are the converses of the above statements true?
Step-by-step solution
Idea: Every pair satisfies exactly one of the three ratio conditions, and each condition leads to a different outcome. So if we know the outcome, we can work backwards to the condition.
- The three statements are: (1) a1a2 ≠ b1b2 ⇒ unique solution; (2) a1a2 = b1b2 = c1c2 ⇒ infinitely many; (3) a1a2 = b1b2 ≠ c1c2 ⇒ no solution. The converses swap “if” and “then”.½ mark
- Any pair falls under exactly one condition: either a1a2 ≠ b1b2, or they are equal and then c1c2 is either equal or not. So the three conditions cover all cases with no overlap.1 mark
- Converse of (1): suppose a pair has a unique solution. It cannot satisfy condition (2) (that would give infinitely many) or (3) (that would give none). So it satisfies (1): a1a2 ≠ b1b2 ✓.½ mark
- The same argument works for the others: infinitely many solutions rules out (1) and (3), so (2) holds; no solution rules out (1) and (2), so (3) holds. All three converses are true.1 mark
Check: x + y = 2, x − y = 0 has the unique solution (1, 1), and indeed 11 ≠ 1−1 ✓.
Answer to write in the exam
Every pair satisfies exactly one of: a1a2 ≠ b1b2; a1a2 = b1b2 = c1c2; a1a2 = b1b2 ≠ c1c2
These give unique, infinitely many, no solution respectively
Unique solution ⇒ not the 2nd or 3rd condition ⇒ a1a2 ≠ b1b2
Similarly for the other two
∴ All the converses are true
Common mistakes that cost marks
- Assuming a converse is true because the statement is true. Here it is true, but only because the cases cover everything and do not overlap; this needs to be said.
- Forgetting the “no overlap” part of the argument.
How this can come in the exam
Assertion (A): If px + 3y = 4 and 2x + 6y = 8 have infinitely many solutions, then p = 1.
Reason (R): Infinitely many solutions ⇒ a1a2 = b1b2 = c1c2.
- 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.
p2 = 36 = 48 = 12 ⇒ p = 1; R (a converse) is what we used.
Try one yourself
A pair 2x + ky = 3, 4x + 6y = 5 has no solution. Find k.
Show answer
No solution ⇒ 24 = k6 ⇒ k = 3 (and 35 ≠ 12 ✓).
More questions like this
- 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?
- Solve the pair of linear equations x + 3y = 6 and 2x − 3y = 12 graphically. Comment on the nature of the solution.
- Determine if the lines x + 2y − 4 = 0 and 2x + 4y − 12 = 0 intersect, coincide or are parallel to each other. Then verify your answer by graphing the equations.
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