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Given the equations 3x + y =17, 5y + z =14 and 3x +5z = 41, what is the value of the sum x + y + z ?

 Nov 4, 2019
 #1
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Given the equations 3x + y =17, 5y + z =14 and 3x +5z = 41, what is the value of the sum x + y + z ?

 

\(\begin{bmatrix} 3 && 1 && 0 &&|&&17 \\ 0 && 5 && 1 &&|&&14\\ 3 && 0 && 5 &&| && 41\\ \end{bmatrix}\\~\\ \begin{bmatrix} 3 && 1 && 0 &&|&&17 \\ 0 && 5 && 1 &&|&&14\\ 0 && -1 && 5 &&| && 24\\ \end{bmatrix}\\~\\ \begin{bmatrix} 3 && 1 && 0 &&|&&17 \\ 0 && 5 && 1 &&|&&14\\ 0 && -5 && 25 &&| && 120\\ \end{bmatrix}\\~\\ \begin{bmatrix} 3 && 1 && 0 &&|&&17 \\ 0 && 5 && 1 &&|&&14\\ 0 && 0 && 26 &&| && 134\\ \end{bmatrix}\\~\\\)

26Z=134

Z=134/26

Z=67/13

 

5y+Z=14

5y=14-134/26

y=(14-134/26)/5 

y=23/13

 

3x+y=17

3x+23/13=17

x=(17-23/13)/3

x= 66/13

 

x+y+z = 66/13 + 23/13 + 67/13 = 12

 Nov 7, 2019
 #2
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Add the three equations together and you get

\(\displaystyle 6x+6y+6z=(6(x+y+z))=72\\\text{so}\\x+y+z =12.\)

.
 Nov 7, 2019

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