Question

Solve that the following pair of linear equations are consistent or inconsistent Sx-3y11, 10 x 6--22​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{5x-3y=11}$

$\mathsf{10x-6y=22}$

$\underline{\textsf{To find:}}$

$\textsf{The solution of the system of equations}$

$\underline{\textsf{Solution:}}$

$\textsf{Consider,}$

$\mathsf{5x-3y=11}$

$\mathsf{10x-6y=22}$

$\textsf{Here,}$

$\mathsf{a_1=5,\,b_1=-3,\,c_1=11}$

$\mathsf{a_2=10,\,b_2=-6,\,c_2=22}$

$\mathsf{a_1\,b_2-a_2\,b_1=-30+30=0}$

$\textsf{But,}$

$\mathsf{\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}=\dfrac{1}{2}}$

$\implies\textsf{Coefficients are proportional}$

$\therefore\textsf{The given system of equations is consistent and}$

$\textsf{has infinitely many solutions}$

$\textsf{In this case,system is reduced to a single equation}$

$\mathsf{5x-3y=11}$

$\textsf{Take,}\;\mathsf{y=k}$

$\mathsf{5x=3k+11}$

$\implies\mathsf{x=\dfrac{3k+11}{5}}$

$\underline{\textsf{Answer:}}$

$\textsf{The solution is}\;$

$\mathsf{x=\dfrac{3k+11}{5},\;y=k}\;\textsf{where}\;\mathsf{k{\in}R}$

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

SOLVE USING DETERMINANT METHOD

A given set of equations are said to be Consistent if

Either one of the below is true

1. Unique Solution when

$\Delta \ne \: 0$

2. Infinite number of solutions when

$\Delta = \Delta_1 = \Delta_2 = 0$

$\: \sf{Where \: \: \Delta \: , \: \Delta_1 \: , \: \Delta_2\: } have \: usual \: notations$

CALCULATION

The given set of equations are

$5x - 3y = 11 \: \: \: \: \: ........(1)$

$10x - 6y = 22 \: \: \: .........(2)$

Which can be rewritten in AX = B form as below :

$\displaystyle\begin{pmatrix} 5 & - 3\\ 10 & - 6 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} 11\\ 22 \end{pmatrix}$

Here

$\Delta = \displaystyle\begin{vmatrix} 5 & - 3\\ 10 & - 6 \end{vmatrix} = - 30 + 30 = 0$

$\Delta_1 = \displaystyle\begin{vmatrix} 5 & 11\\ 10 & 22 \end{vmatrix} = 110 - 110 = 0$

$\Delta_2 = \displaystyle\begin{vmatrix} 11 & - 3\\ 22 & - 6 \end{vmatrix} = - 66 + 66 = 0$

So

$\Delta = \Delta_1 = \Delta_2 = 0$

RESULT

Hence the given set of equations are Consistent

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ADDITIONAL INFORMATION

The given set of equations have infinite number of solutions