In the equations given below, m and n are unknown constants: 2mx + 3y = 7; 4x + ny = −10. If (2, −1) is the solution of both equations, find the values of m and n.
Answer: m = 52, n = 18.
Step-by-step solution
Given: 2mx + 3y = 7 and 4x + ny = −10; x = 2, y = −1 satisfies both
To find: m and n
To find: m and n
Idea: Substitute x = 2 and y = −1 into each equation. Each becomes an equation in one unknown constant.
- First equation: 2m(2) + 3(−1) = 7, i.e. 4m − 3 = 7.½ mark
- 4m = 10, so m = 52 (= 2.5).1 mark
- Second equation: 4(2) + n(−1) = −10, i.e. 8 − n = −10.½ mark
- −n = −18, so n = 18.1 mark
m = 5/2 and n = 18.
Check: m = 52 makes the first equation 5x + 3y = 7: 10 − 3 = 7 ✓. n = 18 makes the second 4x + 18y = −10: 8 − 18 = −10 ✓.
Answer to write in the exam
Put x = 2, y = −1 in 2mx + 3y = 7:
4m − 3 = 7 ⇒ 4m = 10 ⇒ m = 52
Put x = 2, y = −1 in 4x + ny = −10:
8 − n = −10 ⇒ n = 18
∴ m = 52, n = 18
Common mistakes that cost marks
- Writing 3(−1) as +3, giving 4m = 4 and m = 1.
- From 8 − n = −10 getting n = −18 or n = 2. Move 8 across: −n = −18, so n = 18.
How this can come in the exam
MCQ (1 mark)
If x = 3, y = 2 satisfies 2x − ky = 2, then k =
- 1
- 2
- −2
- 4
Show answer
(B) 2
6 − 2k = 2 ⇒ 2k = 4 ⇒ k = 2.
Short answer (2 marks)
(−1, 3) is a solution of 5x + py = 4 and of qx − 2y = −9. Find p and q.
Show answer
−5 + 3p = 4 ⇒ p = 3 (1 mark). −q − 6 = −9 ⇒ q = 3 (1 mark).Try one yourself
If (1, −2) satisfies 3mx − y = 8 and 2x + ny = 12, find m and n.
Show answer
3m + 2 = 8 ⇒ m = 2; 2 − 2n = 12 ⇒ n = −5.
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