Solve the following systems of linear equations by Cramer’s rule:

3x + y = 19

3x – y = 23

Given: - Two equations 3x + y = 19 and 3x – y = 23

Tip: - Theorem – Cramer’s Rule

Let there be a system of n simultaneous linear equations and with n unknown given by

and let D_j be the determinant obtained from D after replacing the j^th column by

Then,

provided that D ≠ 0

Now, here we have

3x + y = 19

3x – y = 23

So by comparing with the theorem, let's find D, D₁ and D₂

Solving determinant, expanding along 1^st row

⇒ D = 3( – 1) – (3)(1)

⇒ D = – 3 – 3

⇒ D = – 6

Again,

Solving determinant, expanding along 1^st row

⇒ D₁ = 19( – 1) – (23)(1)

⇒ D₁ = – 19 – 23

⇒ D₁ = – 42

and

Solving determinant, expanding along 1^st row

⇒ D₂ = 3(23) – (19)(3)

⇒ D₂ = 69 – 57

⇒ D₂ = 12

Thus by Cramer’s Rule, we have

⇒

⇒

⇒ x = 7

and

⇒

⇒

⇒ y = – 2