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Column Method: 2x − 3xy + 5y

x
++ 2xy − 3y
− − +
x − 5xy + 8y

Thus, the required number is x − 5xy + 8y.
Example 25: What must be subtracted from (x + y + 1) to obtain (x − y)?

Solution: Required number = (x + y + 1) – (x − y) = x + y + 1 – x + y = 2y + 1
Thus, 2y + 1 must be subtracted from (x + y + 1) to obtain (x − y).
Example 26: What must be added to (−2x + 6xy + 7) to obtain (3x − xy + 1)?

Solution: Required number = (3x − xy + 1) – (−2x + 6xy + 7)
= 3x − xy + 1 + 2x − 6xy − 7

= (3 + 2)x + (−1 − 6)xy + (1 − 7)
= 5x − 7xy − 6

Thus, (5x − 7xy − 6 ) must be added to (−2x + 6xy + 7) to obtain (3x − xy + 1).

Practice Time 5F

1. Add the following:

(a) 2x, −3x and 7x (b) 3a, 2a, −4a (c) 4pq, 12pq, 32pq (d) 13xy, −12xy, 16xy
2. Subtract the following:

(a) 3xy from 7xy (b) 3xy from −7xy (c) −3xy from 7xy (d) −3xy from −7xy.

3. Add the following by column and horizontal method:
(a) 2x − 3y, 4y − z and 2z + x (b) 3ab + bc − ca, 4ab − 5bc and 3ca − 2ab

(c) 3x − 2xy + 1, 2xy − 5x + 2 and 4x − 3xy + 7 (d) 4pq − 2qr + rp, 3qr − 2rp + pq and pq − 3qr

4. Subtract the following by column and horizontal method:
(a) 7a + 8b − c from 5a − b + 3c (b) 2x − 3y from z − x

(c) 3pq − 2p + q from 4pq + 7p − 2p (d) 11xy − 5y + 2x from 15xy + y – x

5. Subtract 2x + y from the sum of (x − 2xy + y) and (3x + 2xy − 2y).
6. What must be added to 3mn – 2m + 1 to get 5mn + n − 2?

7. What must be subtracted from 2p + q − 1 to get p – 2q + 6?
8. If A = x − xy + y, B = 1 + xy and C = y − x − xy, find:

(i) A + B + C (ii) A + B − C
9. Three sides of a triangle are (2x + y) units, (3x − y) units and (x + 3y) units. Find its perimeter.

10. The perimeter of a triangle is (3x + 4y) units. If its two sides are (x + 2y) units and (2x + y) units, find
the third side.

121 Introduction to Algebra