Question

Solve by Cramer’s rule, 3x – 4y = 10, 4x + 3y = 5

Answer 1

The given equations are

3x – 4y = 10     ......(i)

4x + 3y = 5  .....(ii)

Equations (i) and (ii) are in ax + by = c form.

Comparing the given equations with a₁x + b₁y = c₁ and a₂x + b₂y = c₂, we get

a₁ = 3, b₁ = – 4, c₁ = 10 and

a₂ = 4, b₂ = 3, c₂ = 5

∴ D = `|("a"_1, "b"_1),("a"_2, "b"_2)|`

= `|(3, -4),(4, 3)|`

= (3 × 3) – (– 4 × 4)

= 9 − (−16)

= 9 + 16

= 25 ≠ 0

D_x = `|("c"_1, "b"_1),("c"_2, "b"_2)|`

= `|(10, -4),(5, 3)|`

= (10 × 3) – (– 4 × 5)

= 30 − (−20)

= 30 + 20

= 50

D_y = `|("a"_1, "c"_1),("a"_2, "c"_2)|`

= `|(3, 10),(4, 5)|`

= (3 × 5) – (10 × 4)

= 15 – 40

= – 25

∴ By Cramer’s rule, we get

x = `("D"_x)/"D"` and y = `("D"_y)/"D"`

∴ x = `50/25` and y = `(-25)/25`

∴ x = 2 and y = –1

∴ (x, y) = (2, –1) is the solution of the given equations.