Question

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

${\displaystyle \frac{3}{2}+2y=12,\frac{2}{x}+3y}$ = 13

Answer 1

Put ${\displaystyle \frac{1}{x}}$ = a

⇒ 3a + 2y = 12

2a + 3y = 13

Writing the above equations in matrix form we get

${\displaystyle \left[\begin{array}{cc}3& 2\\ 2& 3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}12& 13\end{array}\right]}$

(i.e) AX = B

Now |A| = Δ = ${\displaystyle \left|\begin{array}{cc}3& 2\\ 2& 3\end{array}\right|}$ = 9 – 4 = 5 ≠ 0

Δ_a = ${\displaystyle \left|\begin{array}{cc}12& 2\\ 13& 3\end{array}\right|}$ = 36 – 26 = 10

Δ_y = ${\displaystyle \left|\begin{array}{cc}3& 12\\ 2& 13\end{array}\right|}$ = 39 – 24 = 15

a = ${\displaystyle \frac{{\Delta }_{\text{a}}}{\Delta }=\frac{10}{5}}$ = 2

y = ${\displaystyle \frac{{\Delta }_y}{\Delta }=\frac{15}{5}}$ = – 3

∴ a = 2, y = 3 but ${\displaystyle \frac{1}{x}}$ = a

⇒ x = ${\displaystyle \frac{1}{\text{a}}=\frac{1}{2}}$ and y = 3

∴ x = ${\displaystyle \frac{1}{2}}$, y = 3