Question

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

9x + 5y = 10 

3y – 2x = 8

Answer 1

Given : - 

Two equations 9x + 5y = 10 and 3y – 2x = 8

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

\(\begin{vmatrix} b_1 \\[0.3em] b_2 \\[0.3em]: \\[0.3em] b_n \end{vmatrix}\)

Then,

x₁ = \(\frac{D_1}{D},\) x₂ =\(\frac{D_2}{D},\)....,x_n = \(\frac{D_n}{D}\)

provided that D ≠ 0

Now, here we have

9x + 5y = 10

3y – 2x = 8

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

⇒ D = \(\begin{vmatrix} 9&5 \\[0.3em] -2 &3 \\[0.3em] \end{vmatrix}\)

Solving determinant, expanding along 1^st row 

⇒ D = 3(9) – (5)( – 2) 

⇒ D = 27 + 10 

⇒ D = 37 

Again,

⇒ D₁ = \(\begin{vmatrix} 10&5 \\[0.3em] 8 &3 \\[0.3em] \end{vmatrix}\)

Solving determinant, expanding along 1st row 

⇒ D₁ = 10(3) – (8)(5) 

⇒ D₁ = 30 – 40 

⇒ D₁ = – 10 

And,

⇒ D₂ = \(\begin{vmatrix} 9&10 \\[0.3em] -2 &8 \\[0.3em] \end{vmatrix}\)

Solving determinant, expanding along 1st row 

⇒ D₂ = 9(8) – (10)( – 2) 

⇒ D₂ = 72 + 20 

⇒ D₂ = 92

Thus by Cramer’s Rule, we have