Question

can we find a unique solution for the following simultaneous equations why or why not x+y=10,5x+5y=50​

Answer 1

Question :-

• Can we find a unique solution for the following simultaneous equations why or why not x+y=10,5x+5y=50.

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$\begin{gathered}\Large{\bold{\pink{\underline{CaLcUlAtIoN\::}}}}\end{gathered}$

$\bf \:\large \red{Concept \: used} ✍$

Let us consider two linear equations

$\bf \:  a_1x + b_1y + c_1 = 0 \: and \: a_2x + b_2y + c_2 = 0$

then consistency or inconsistency depends upon the following situations

$\bf \:\small \red{Case :- 1.} ✍$

$\bf \:If\bf \:\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

then system of equations is consistent having infinitely many solutions

$\bf \:\small \red{Case :- 2.} ✍$

$\bf \:If\bf \:\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} ≠ \dfrac{c_1}{c_2}$

then system of equations is inconsistent having no solution.

$\bf \:\small \red{Case :- 3.} ✍$

$\bf \:If\bf \:\dfrac{a_1}{a_2} ≠ \dfrac{b_1}{b_2}$

then system of equations is consistent having unique solution.

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$\large \red{\bf \:Solution :- } ✍$

For the given system of linear equations

$\bf \:x+y=10, \: 5x+5y=50$

$\bf \:\bf \:\dfrac{a_1}{a_2} = \dfrac{1}{5}$

$\bf \ \: \dfrac{b_1}{b_2} = \dfrac{1}{5}$

$\bf \:\dfrac{c_1}{c_2} = \dfrac{10}{50} = \dfrac{1}{5}$

So, we concluded that

$\bf \:\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} = \dfrac{1}{5}$

So, system of equations is consistent having infinitely many solutions.

$\begin{gathered}\bf\red{So, \: not \: able \:to \: find \: unique \: {sol}^{n} } .\\ \end{gathered}$

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