Show that Each One of the Following Systems of Linear Equation is Inconsistent: 2x + 3y = 5 6x + 9y = 10 - Mathematics

प्रश्न

Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10

उत्तर

The given system of equations can be expressed as follows:
$A X = B$
Here,
$A = [\begin{matrix} 2 & 3 \\ 6 & 9 \end{matrix}], X = (\binom{x}{y}) and B = (\binom{5}{10})$
Now,
$| A | = | \begin{matrix} 2 & 3 \\ 6 & 9 \end{matrix} |$
$= (18 - 18)$
$= 0$
${Let C}_{i j} {be the cofactors of the elements a}_{i j} in A = [a_{i j}] . Then,$
$C_{11} = {(- 1)}^{1 + 1} (9) = 9, C_{12} = {(- 1)}^{1 + 2} (6) = - 6$
$C_{21} = {(- 1)}^{2 + 1} (3) = - 3, C_{22} = {(- 1)}^{2 + 2} (2) = 2$
$a d j A = {[\begin{matrix} 9 & - 6 \\ - 3 & 2 \end{matrix}]}^{T}$
$= [\begin{matrix} 9 & - 3 \\ - 6 & 2 \end{matrix}]$
$(a d j A) B = [\begin{matrix} 9 & - 3 \\ - 6 & 2 \end{matrix}] (\binom{5}{10})$
$= (\binom{45 - 30}{- 30 + 20})$
$= (\binom{15}{- 10}) \neq 0$
Hence, the given system of equations is inconsistent.

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Applications of Determinants and Matrices

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पाठ 8: Solution of Simultaneous Linear Equations - Exercise 8.1 [पृष्ठ १५]

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पाठ 8 Solution of Simultaneous Linear Equations
Exercise 8.1 | Q 4.2 | पृष्ठ १५

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Show that Each One of the Following Systems of Linear Equation is Inconsistent: 2x + 3y = 5 6x + 9y = 10 - Mathematics

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Show that Each One of the Following Systems of Linear Equation is Inconsistent: 2x + 3y = 5 6x + 9y = 10 - Mathematics

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