Solutions of a linear equation · 2 marks

Find any two solutions for each of the following equations:

(i) 7x − 3y = 21
(ii) 2x + 3y = 5

Answer: (i) (0, −7) and (3, 0) (ii) (1, 1) and (4, −1). (Many other answers are possible.)

Step-by-step solution

Idea: Choose a value for one variable and solve the resulting one-variable equation for the other. Pick values that give whole numbers when you can.

(i) 7x − 3y = 21

x = 0: −3y = 21, so y = −7 → (0, −7).½ mark
y = 0: 7x = 21, so x = 3 → (3, 0). (Another: x = 6 gives −3y = −21, y = 7 → (6, 7).)½ mark

(0, −7) and (3, 0)

(ii) 2x + 3y = 5

x = 1: 2 + 3y = 5, so y = 1 → (1, 1).½ mark
x = 4: 8 + 3y = 5, so y = −1 → (4, −1). (With y = 0 you get (52, 0), also correct.)½ mark

(1, 1) and (4, −1)

(i) (0, −7) and (3, 0); (ii) (1, 1) and (4, −1). Any pairs that satisfy the equations are correct.

Check: (i) 7(3) − 3(0) = 21 ✓, 7(0) − 3(−7) = 21 ✓. (ii) 2 + 3 = 5 ✓, 8 − 3 = 5 ✓.

Answer to write in the exam

(i)

x = 0: −3y = 21 ⇒ y = −7

y = 0: 7x = 21 ⇒ x = 3

∴ Two solutions: (0, −7), (3, 0)

(ii)

x = 1: 2 + 3y = 5 ⇒ y = 1

x = 4: 8 + 3y = 5 ⇒ y = −1

∴ Two solutions: (1, 1), (4, −1)

Common mistakes that cost marks

In (i), getting y = 7 from −3y = 21: dividing by −3 gives −7.
Giving only one solution, or two pairs that are the same point written differently.

How this can come in the exam

MCQ (1 mark)

Which of these is a solution of 4x − 5y = 3?

(2, 1)
(1, 2)
(−2, 1)
(3, 2)

Show answer

(A) (2, 1)
4(2) − 5(1) = 3 ✓.

Try one yourself

Find two solutions of 5x − 2y = 10.