Learnify Academy is a tuition centre in Bahrain. Classes are for students in Bahrain only.Tuition classes in Bahrain only

Solving equations · 2 marks

If x = y, then a + x = a + y, where x, y and a are any three numbers.
This proposition and its converse are routinely used while solving equations.

Answer: Converse: If a + x = a + y, then x = y. Both are true.

Step-by-step solution

To find: Frame the converse of this proposition. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement.

Idea: Adding the same number to two equal numbers keeps them equal; and subtracting the same number a from both sides undoes the addition, so equal sums with the same a come from equal numbers.

  1. Converse: If a + x = a + y, then x = y.½ mark
  2. Proposition — true. x and y are the same number, so adding the same number a to each gives the same result: a + x = a + y.½ mark
  3. Converse — true. Subtract a from both sides of a + x = a + y (that is, use the proposition with −a): (a + x) − a = (a + y) − a, so x = y.½ mark
  4. Both are true. For example, x − 5 = 12 becomes x = 17 by adding 5 to both sides (proposition), and 5 + x = 5 + 9 gives x = 9 by cancelling 5 (converse).½ mark
Converse: If a + x = a + y, then x = y. Both the proposition and the converse are true.

Answer to write in the exam

Converse: If a + x = a + y, then x = y.

Proposition: x = y ⇒ adding a to both: a + x = a + y. True.

Converse: subtract a from both sides: (a + x) − a = (a + y) − a ⇒ x = y. True.

∴ Both the proposition and its converse are true.

Common mistakes that cost marks

  • Thinking the converse fails ‘because a could be anything’. a is the same number on both sides, so it can always be subtracted away.
  • Carrying the idea over to multiplication: ‘If ax = ay, then x = y‘ is false when a = 0. Addition has no such exception.

How this can come in the exam

Assertion–Reason (1 mark)

Assertion (A): If 7 + x = 7 + 12, then x = 12.
Reason (R): If a + x = a + y, then x = y, for any numbers a, x, y.

  1. Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
  2. Both Assertion (A) and Reason (R) are true but R is not the correct explanation of A.
  3. Assertion (A) is true but Reason (R) is false.
  4. 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.
A is R with a = 7 and y = 12, so R is the correct explanation.

MCQ (1 mark)

Which statement is NOT true for all numbers a, x, y?

  1. If x = y, then x − a = y − a.
  2. If x − a = y − a, then x = y.
  3. If ax = ay, then x = y.
  4. If x = y, then ax = ay.
Show answer

(C) If ax = ay, then x = y.
With a = 0: 0 × 3 = 0 × 5, but 3 ≠ 5. The other three always hold.

Try one yourself

Write the converse of ‘If x = y, then x − 4 = y − 4′. Is it true?

Show answer

Converse: ‘If x − 4 = y − 4, then x = y.’ True: add 4 to both sides.

More questions like this

All Mathematical reasoning questions · All maths questions

← Given any ∆ABC, let us bisect the angles at B and C. The bisectors meeIf a and b are perfect squares, then ab is a perfect square. →