What if we had expressed y in terms of x, i.e., take Equation (2) and write y = 12(3 − x). Would we still get the same solution?
Step-by-step solution
Idea: A pair of equations has its solution whatever route we take. Choosing which variable to isolate only changes the arithmetic, not the answer.
- The pair is 7x − 15y = 2 … (1) and x + 2y = 3 … (2). From (2): y = 3 − x2.½ mark
- Substitute in (1): 7x − 15(3 − x2) = 2. Multiply by 2: 14x − 45 + 15x = 4 ⇒ 29x = 49 ⇒ x = 4929.1 mark
- y = ½(3 − 4929) = ½ × 3829 = 1929.1 mark
- This is the same solution as before. Yes, the answer does not depend on which variable we express first; here isolating x was a little easier because it avoided the fraction ½.½ mark
Check: Same check as before: 7(4929) − 15(1929) = 2 ✓ and 4929 + 2(1929) = 3 ✓.
Answer to write in the exam
From (2): y = 3 − x2
Put in (1): 7x − 15(3 − x)2 = 2
14x − 45 + 15x = 4 ⇒ 29x = 49 ⇒ x = 4929
y = 12(3 − 4929) = 1929
∴ Same solution: x = 4929, y = 1929
Common mistakes that cost marks
- Multiplying only some terms by 2 when clearing the fraction: every term, including the right-hand 2, must be doubled.
- Writing −15 × (3 − x) as −45 − 15x. The minus times minus gives +15x.
How this can come in the exam
Assertion (A): Solving x + y = 5, x − y = 1 by isolating x or by isolating y gives the same answer.
Reason (R): A pair of linear equations with a unique solution has only one common 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.
Both routes give (3, 2); R explains why they must agree.
Try one yourself
Solve 2x + y = 8, x − y = 1 twice: once isolating y in the first, once isolating x in the second.
Show answer
Both give x = 3, y = 2.
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