Please solve RD Sharma class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 4 subquestion (i) maths textbook solution

Inconsistent

Given:

$2x+5y=7 \: \: ,\: \: 6x+15y=13$

Hint:

Inconsistent means two or more equations that are impossible to solve based on using one set of values for variables.

Solution:

The given system of equations can be expressed as follows

$A X=B \\ Here, \\ A=\left[\begin{array}{cc}2 & 5 \\ 6 & 15\end{array}\right] \quad, \quad X=\left[\begin{array}{l}x \\ y\end{array}\right] \quad and \quad B=\left[\begin{array}{c}7 \\ 13\end{array}\right] \\ Now, \\ |A|=\left|\begin{array}{cc}2 & 5 \\ 6 & 15\end{array}\right|=|30-30|=0$

$C_{i j}\; be\; the\; co-\! f\! actor\; o\! f\; the\; elements\; a_{i j}\; in \; A=\left[a_{i j}\right].\; Then,$

\begin{aligned} &C_{11}=-1^{1+1}(15)=15 \quad, \quad C_{12}=-1^{1+2}(6)=-6 \\ &C_{21}=-1^{2+1}(5)=-5 \quad, \quad C_{22}=-1^{2+2}(2)=2 \\ &(a d j A) B=\left[\begin{array}{cc} 15 & -5 \\ -6 & 2 \end{array}\right]\left[\begin{array}{c} 7 \\ 13 \end{array}\right]=\left[\begin{array}{c} 105-65 \\ -42+26 \end{array}\right]=\left[\begin{array}{c} 40 \\ -16 \end{array}\right] \neq 0 \end{aligned}

Hence, the given system of equation is inconsistent.