Respuesta :
Answer:
the answer is C: ( -1,2,4)
Step-by-step explanation:
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The solution of given system of equations is [tex]\bold{x=\frac{-5}{3},~y=\frac{43}{21},~z=\frac{80}{21}}[/tex]
What is system of equations?
"It is a finite set of equations for which we find the common solution."
What is row-echelon form?
"It is a simplified equivalent version of a matrix which has been reduced row by row."
What is reduced row-echelon form?
"It is a type of matrix used to solve systems of linear equations."
What is augmented matrix for system of equations?
"It is a matrix of numbers in which each row represents the constants from one equation and each column represents all the coefficients for a single variable."
For given question,
we have been given a system of equations
2x + 2y + 4z = 16
5x - 2y + 3z = -1
x + 2y - 3z = -9
The augmented matrix for above system of equations would be,
[tex]\begin{bmatrix}2 & 2 & 4 & 16 \\5 & -2 & 3 & -1 \\1 & 2 & -3 & -9\end{bmatrix}[/tex]
Now, we find the reduced row-echelon form of the augmented matrix.
[tex]=\begin{bmatrix}2 & 2 & 4 & 16 \\5 & -2 & 3 & -1 \\1 & 2 & -3 & -9\end{bmatrix}\\\\\\=\begin{bmatrix}2 & 2 & 4 & 16 \\0 & -7 & -7 & -41 \\1 & 2 & -3 & -9\end{bmatrix} ~~~~~~~~~.................(R2\rightarrow R2-\frac{5}{2} R1)\\\\\\=\begin{bmatrix}2 & 2 & 4 & 16 \\0 & -7 & -7 & -41 \\0 & 1 & -5 & -17\end{bmatrix}~~~~~~~~~...................(R3\rightarrow R3-\frac{1}{2} R1)[/tex]
[tex]=\begin{bmatrix}2 & 2 & 4 & 16 \\0 & -7 & -7 & -41 \\0 & 0 & -6 & \frac{-160}{7}\end{bmatrix}~~~~~~~~......................(R3\rightarrow R3-(\frac{-1}{7} )R2)[/tex]
So, we get system of equations,
[tex](2\times x) +(2\times y) +(4\times z) = 16 \\\\(-7\times y) -(7\times z) = -41 \\\\(-6\times z) = \frac{-160}{7}[/tex]
After solving above system of equations we have,
[tex]x=\frac{-5}{3},~y=\frac{43}{21},~z=\frac{80}{21}[/tex]
Therefore, the solution of given system of equations is,
[tex]\bold{x=\frac{-5}{3},~y=\frac{43}{21},~z=\frac{80}{21}}[/tex]
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