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# How do you write the augmented matrix for the system of linear equations $7x - 5y + z = 13,\,19x = 8z = 10$?

Last updated date: 28th Feb 2024
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Hint:Start by computing the values for $(f - g)(x)$. Then one by one substitute the values of the terms $f(x)$ and $g(x)$. Then further open the brackets and combine all the like together. Finally evaluate the value of the function $(f - g)(3)$.

First we will start off by mentioning both the equations separately
$7x - 5y + z = 13$….. (1)
$\,\,\,\,\,19x = 8z = 10$ …… (2)
Now here, we will split equation (2) into two equations.
$7x - 5y + z = 13$….. (1)
$\,\,\,\,\,19x\,\,\,\,\,\,\,\, = 10$ ….... (2)
$+ 8x = 10$…… (3)
Now we will write the augmented matrix.
$\left[ {\begin{array}{*{20}{c}} 7&{ - 5}&{1|\,\,\,13} \\ {19}&0&{0|\,\,\,10} \\ 0&0&{8|\,\,\,10} \end{array}} \right]$
We will start by applying the operation ${R_3} \div 8$ and ${R_2} \div 19$.
Hence, the matrix will become,
$\left[ {\begin{array}{*{20}{c}} 7&{ - 5}&{1|\,\,\,13} \\ 1&0&{0|\,\,\,\dfrac{{10}}{{19}}} \\ 0&0&{1|\,\,\,\dfrac{{10}}{8}} \end{array}} \right]$
We will now apply the operation ${R_1} - 7{R_2}$.
Hence, the matrix will become,
$\left[ {\begin{array}{*{20}{c}} 0&{ - 5}&{1\,\,\,\,\,|\,\,\dfrac{{237}}{{19}}} \\ 1&0&{0|\,\,\,\dfrac{{10}}{{19}}} \\ 0&0&{1|\,\,\,\dfrac{{10}}{8}} \end{array}} \right]$
We will now apply the operation ${R_1} \div ( - 5)$.
Hence, the matrix will become,
$\left[ {\begin{array}{*{20}{c}} 0&1&{\dfrac{{ - 1}}{5}\,\,\,\,\,|\,\,\dfrac{{ - 237}}{{95}}} \\ 1&0&{0\,\,\,\,\,|\,\,\,\dfrac{{10}}{{19}}} \\ 0&0&{1\,\,\,\,\,|\,\,\,\dfrac{{10}}{8}} \end{array}} \right]$
We will now apply the operation ${R_1} + \dfrac{1}{5}{R_3}$.
Hence, the matrix will become,
$\left[ {\begin{array}{*{20}{c}} 0&1&{0\,\,\,\,\,\,\,\,\,\,\,\,\dfrac{{ - 667}}{{380}}} \\ 1&0&{0\,\,\,\,\,|\,\,\,\dfrac{{10}}{{19}}} \\ 0&0&{1\,\,\,\,\,|\,\,\,\dfrac{{10}}{8}} \end{array}} \right]$
$\left[ {\begin{array}{*{20}{c}} 1&0&{0\,\,\,|\,\,\,\dfrac{{10}}{{19}}} \\ 0&1&{\,\,\,0\,\,\,|\,\,\,\dfrac{{ - 667}}{{380}}} \\ 0&0&{1\,\,\,|\,\,\,\dfrac{5}{4}} \end{array}} \right]$