+152 a) Four positive integers \(w, x,y,and \) \(z \) satisfy \(wx + w + x &= 524, xy + x + y &= 146, yz + y + z &= 104,\) and wxyz= 8! what is w-z??? The answer is 10 btw but its to help on part b.
b) In Part a of this problem, you found the values of integers that satisfied the system of equations. In this part of the problem you will write up a full solution that describes how to solve this problem.
Four positive integers \(w, x,y,and \) \(z \) satisfy \(wx + w + x &= 524, xy + x + y &= 146, yz + y + z &= 104,\) and wxyz= 8! What are w, , y, and z???
+26367 Four positive integers \(w\), \(x\), \(y\),and \(z\) satisfy \(wx + w + x = 524\), \(xy + x + y = 146\), \(yz + y + z = 104\), and \(wxyz= 8!\).
What are \(w\), \(x\), \(y\), and \(z\) ?
Solution by substitution:
\(\begin{array}{|rcll|} \hline wx + w + x &=& 524 \\ x(w+1)+w &=& 524 \\ x(w+1) &=& 524-w \\ \mathbf{x} &=& \mathbf{\dfrac{524-w} {w+1}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline xy + x + y &=& 146 \\ y(x+1) + x &=& 146 \\ y(x+1) &=& 146-x \\ \mathbf{y} &=& \mathbf{\dfrac{146-x} {x+1}} \\\\ y &=& \dfrac{146-\dfrac{524-w} {w+1}} {\dfrac{524-w} {w+1}+1} \\ \ldots \\ \mathbf{y} &=& \mathbf{\dfrac{147w-378}{525} } \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline yz + y + z &=& 104 \\ z(y+1) + x &=& 146 \\ z(y+1) &=& 104-y \\ \mathbf{z} &=& \mathbf{\dfrac{104-y} {y+1}} \\ z &=& \dfrac{104-\dfrac{147w-378}{525}} {\dfrac{147w-378}{525}+1} \\ \ldots \\ \mathbf{z} &=& \mathbf{\dfrac{54978-147w}{147(w+1)} } \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{wxyz} &=& \mathbf{8!} \\ wxyz &=& 40320 \\ 40320 &=& w \left(\dfrac{524-w} {w+1}\right) \left(\dfrac{147w-378}{525} \right) \left(\dfrac{54978-147w}{147(w+1)} \right) \\ 40320 &=& \dfrac{w(524-w)(147w-378)(54978-147w)} {77175(w+1)^2} \\ 3111696000 (w+1)^2 &=& w(524-w)(147w-378 )(54978-147w ) \\ \ldots \\ \hline \end{array}\\ \begin{array}{|lrcll|} \hline & -21609 w^4 + 19460448 w^3 - 1173047652 w^2 + 17112994416 w + 3111696000 &=& 0 \\ \hline \end{array}\)
\(\begin{array}{|lrcll|} \hline \text{Solutions: }\\ &\mathbf{ w } &=& \mathbf{24}\ \checkmark \qquad \text{integer} \\ & w&=&0.17961 \\ &w&=&39.918 \\ &w&=&36.83 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{x} &=& \mathbf{\dfrac{524-w} {w+1}} \\ x &=& \dfrac{524-24} {24+1} \\ x &=& \dfrac{500} {25} \\ \mathbf{x} &=& \mathbf{20} \\\\ \mathbf{y} &=& \mathbf{\dfrac{146-x} {x+1}} \\ y &=& \dfrac{146-20} {20+1} \\ y &=& \dfrac{126} {21} \\ \mathbf{y} &=& \mathbf{6} \\\\ \mathbf{z} &=& \mathbf{\dfrac{104-y} {y+1}} \\ z &=& \dfrac{104-6} {6+1} \\ z &=& \dfrac{98} {7} \\ \mathbf{z} &=& \mathbf{14} \\ \hline \end{array}\)
