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HELPPP!!!!

Solve this system of equations using the inverse of a matrix.
-x +y = 8
4x + 3y = 12
x - 7y + z = 15

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The Solution of -x +y = 84, x + 3y = 12, x - 7y + z = 15 be $$\; \left[\begin{array}{ccc}x\\y\\z\end{array}\right] \; =\left[\begin{array}{ccc}-1.76\\6.24\\60.76\end{array}\right]$$

The correct option is (C)

What is inverse of Matrix?

The inverse of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity.

Given equation are:

-x +y = 8

4x + 3y = 12

x - 7y + z = 15

So,

$$\left[\begin{array}{ccc}-1&1&0\\4&3&0\\1&-7&1\end{array}\right] \;*\; \left[\begin{array}{ccc}x\\y\\z\end{array}\right] \; = \; \left[\begin{array}{ccc}8\\12\\15\end{array}\right]$$

then,

$$\; \left[\begin{array}{ccc}x\\y\\z\end{array}\right] \; = \; \left[\begin{array}{ccc}-1&1&0\\4&3&0\\1&-7&1\end{array}\right]^{-1} \; \left[\begin{array}{ccc}8\\12\\15\end{array}\right]$$

Now, Calculating the inverse of Matrix we have,

$$\left[\begin{array}{ccc}-1&1&0\\4&3&0\\1&-7&1\end{array}\right]^{-1} \;$$ = $$\left[\begin{array}{ccc}-3/7&1/7&0\\4/7&1/7&0\\31/7&6/7&1\end{array}\right]$$

so, $$\; \left[\begin{array}{ccc}x\\y\\z\end{array}\right] \; = \;\left[\begin{array}{ccc}-3/7&1/7&0\\4/7&1/7&0\\31/7&6/7&1\end{array}\right]\; \left[\begin{array}{ccc}8\\12\\15\end{array}\right]$$

$$\; \left[\begin{array}{ccc}x\\y\\z\end{array}\right] \; =\left[\begin{array}{ccc}-12/7\\44/7\\425/7\end{array}\right]$$

$$\; \left[\begin{array}{ccc}x\\y\\z\end{array}\right] \; =\left[\begin{array}{ccc}-1.76\\6.24\\60.76\end{array}\right]$$

Hence ,$$\; \left[\begin{array}{ccc}x\\y\\z\end{array}\right] \; =\left[\begin{array}{ccc}-1.76\\6.24\\60.76\end{array}\right]$$