Question

solve by Cramer's rule by 3x + 5y=9 2y-5x=16​

Answer 1

$\large\underline{\sf{Solution-}}$

Given pair of linear equation is

$\red{\rm :\longmapsto\:3x + 5y = 9}$

and

$\red{\rm :\longmapsto\: - 5x + 2y = 16}$

So, above equations in matrix form can be represented as

$\purple{\rm :\longmapsto\:\bigg[ \begin{matrix}3&5 \\ - 5&2 \end{matrix} \bigg] \begin{gathered}\sf \left[\begin{array}{c}x\\y\end{array}\right]\end{gathered} \: = \: \begin{gathered}\sf \left[\begin{array}{c}9\\16\end{array}\right]\end{gathered}}$

So, can be represented as

$\pink{\rm :\longmapsto\:AX = B}$

where,

$\purple{\rm :\longmapsto\:A = \:\bigg[ \begin{matrix}3&5 \\ 5&2 \end{matrix} \bigg]}$

$\purple{\rm :\longmapsto\:X = \begin{gathered}\sf \left[\begin{array}{c}x\\y\end{array}\right]\end{gathered}}$

$\purple{\rm :\longmapsto\:B = \begin{gathered}\sf \left[\begin{array}{c}9\\16\end{array}\right]\end{gathered}}$

Now, Consider,

$\rm :\longmapsto\: |A| = \begin{array}{|cc|}\sf 3 &\sf 5 \\ \sf - 5 &\sf 2 \\\end{array} = 6 - ( - 25) = 31$

So, it implies system of equations have unique solution.

So,

$\rm :\longmapsto\: D_1 = \begin{array}{|cc|}\sf 9 &\sf 5 \\ \sf 16 &\sf 2\\\end{array} = 18 - 80 = - 62$

and

$\rm :\longmapsto\: D_2 = \begin{array}{|cc|}\sf 3 &\sf -5 \\ \sf 9 &\sf 16 \\\end{array} = 48 + 45 = 93$

$\bf\implies \:x = \dfrac{D_1}{ |A| } = \dfrac{ - 62}{31} = - 2$

and

$\bf\implies \:y = \dfrac{D_2}{ |A| } = \dfrac{93}{31} = 3$

Verification :

Consider first equation,

$\red{\rm :\longmapsto\:3x + 5y = 9}$

On substituting the values of x and y, we get

$\red{\rm :\longmapsto\:3( - 2) + 5(3) = 9}$

$\red{\rm :\longmapsto\: - 6 + 15= 9}$

$\red{\rm :\longmapsto\: 9= 9}$

Hence, Verified

Consider, Second equation

$\red{\rm :\longmapsto\: - 5x + 2y = 16}$

On substituting the values of x and y, we get

$\red{\rm :\longmapsto\: - 5( - 2) + 2(3) = 16}$

$\red{\rm :\longmapsto\: 10 +6 = 16}$

$\red{\rm :\longmapsto\: 16 = 16}$

Hence, Verified