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$$\begin{cases}x_1+y_1=a\\ x_1-y_2=b\\ y_1-x_2=c\\ x_2+y_2=d \end{cases}$$

How to find a solution for these equations because from elimination and substitution method I end up with 2 equations with totally same variable part on L.H.S like below:

$$\begin{cases}y_1+y_1=A\\ y_1+y_2=B\end{cases}$$

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  • $\begingroup$ So, it must be $A=B$ and there are infinitely many solutions. $\endgroup$
    – mfl
    Feb 10, 2019 at 18:03
  • $\begingroup$ Label the equations (1),(2),(3),(4). Then (1)-(2) gives $y_1+y_2=a-b$ and (3)+(4) gives $y_1+y_2=c+d$. If $a-b\ne c+d$, then the equations are inconsistent and have no solutions. If $a-b=c+d$, then one equation is redundant and you have infinitely many solutions. $\endgroup$
    – almagest
    Feb 10, 2019 at 18:30

1 Answer 1

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Writing the augmented matrix of this linear system in reduced row echelon form, you obtain that either there are no solutions, or the solutions make an affine line in $K^4$. I ordered the unknowns as $(x_1,x_2,y_1,y_2)$:

\begin{align} &\left[\begin{array}{cccc|c} 1&0&1&0&a \\ 1&0&0&-1&b\\ 0&-1& 1&0&c \\ 0&1&0&1&d \end{array}\right]\rightsquigarrow \left[\begin{array}{cccc|c} 1&0&1&0&a \\ 0&1&0&1&d \\ 1&0&0&-1&b\\ 0&-1& 1&0&c \\ \end{array}\right]\rightsquigarrow \left[\begin{array}{cccc|c} 1&0&1&0&a \\ 0&1&0&1&d \\ 0&0&-1&-1&b-a\\ 0&-1& 1&0&c \\ \end{array}\right]\rightsquigarrow \\ \rightsquigarrow&\left[\begin{array}{cccc|c} 1&0&1&0&a \\ 0&1&0&1&d \\ 0&0&-1&-1&b-a\\ 0&0& 1&1&c+d \\ \end{array}\right]\rightsquigarrow \left[\begin{array}{cccc|c} 1&0&0&-1&b \\ 0&1&0&1&d \\ 0&0&1&1&a-b\\ 0&0& 0&0&-a+b+c+d \\ \end{array}\right]. \end{align}

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  • $\begingroup$ +1. Pretty much the standard way to solve any linear system (in fact, the canonical) $\endgroup$ Feb 10, 2019 at 18:33
  • $\begingroup$ your answer is nice and helping and i found another same question as mine because it is required for solving a problem so many people asking math.stackexchange.com/questions/3104871/… $\endgroup$
    – Abhi
    Feb 10, 2019 at 19:22
  • $\begingroup$ if anybody can give more clarification from link in above comment that would be thankfull. $\endgroup$
    – Abhi
    Feb 10, 2019 at 19:24

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