document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209751>Question 1209751</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In this multi-part problem, we will consider this system of simultaneous equations:<BR>\n" );
document.write( "3x + 4y + 30z = -60,<BR>\n" );
document.write( "2xy + 42xz - 16yz = 68,<BR>\n" );
document.write( "5xyz = 56.\r<BR>\n" );
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document.write( "Let a = x/2, b = 5y and c = -4z.\r<BR>\n" );
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document.write( "Determine the monic cubic polynomial in terms of a variable t whose roots are t = a, t = b, and t = c.\r<BR>\n" );
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document.write( "[i]This is a continuation of the problem above.[/i]\r<BR>\n" );
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document.write( "Given that $(x,y,z)$ is a solution to the original system of equations, determine all distinct possible values of $x + y + z$.\r<BR>\n" );
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document.write( "(Suggestion: Using the substitutions in part (a), first determine all possible values of the ordered triple $(a,b,c)$, then determine the possible solutions $(x,y,z)$.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850100 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850100\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the monic cubic polynomial in terms of a variable $t$ whose roots are $t=a$, $t=b$, and $t=c$ be given by<BR>\n" );
document.write( "$$(t-a)(t-b)(t-c) = t^3 - (a+b+c)t^2 + (ab+bc+ca)t - abc = 0$$<BR>\n" );
document.write( "We are given that $a = \frac{x}{2}$, $b = 5y$, and $c = -4z$.<BR>\n" );
document.write( "From the given equations, we have<BR>\n" );
document.write( "\begin{align*} \label{eq:1} 3x + 4y + 30z &= -60 \\ 2xy + 42xz - 16yz &= 68 \\ 5xyz &= 56\end{align*}<BR>\n" );
document.write( "Substituting $x = 2a$, $y = \frac{b}{5}$, and $z = -\frac{c}{4}$ into the given equations, we get<BR>\n" );
document.write( "\begin{align*} 3(2a) + 4\left(\frac{b}{5}\right) + 30\left(-\frac{c}{4}\right) &= -60 \\ 2(2a)\left(\frac{b}{5}\right) + 42(2a)\left(-\frac{c}{4}\right) - 16\left(\frac{b}{5}\right)\left(-\frac{c}{4}\right) &= 68 \\ 5(2a)\left(\frac{b}{5}\right)\left(-\frac{c}{4}\right) &= 56\end{align*}<BR>\n" );
document.write( "Simplifying, we have<BR>\n" );
document.write( "\begin{align*} 6a + \frac{4}{5}b - \frac{15}{2}c &= -60 \\ \frac{4}{5}ab - 21ac + \frac{4}{5}bc &= 68 \\ -abc &= 56\end{align*}<BR>\n" );
document.write( "Multiplying the first equation by $\frac{5}{2}$, we get<BR>\n" );
document.write( "$$15a + 2b - \frac{75}{4}c = -150$$<BR>\n" );
document.write( "Multiplying the second equation by $\frac{5}{4}$, we get<BR>\n" );
document.write( "$$ab - \frac{105}{2}ac + bc = 85$$<BR>\n" );
document.write( "From the third equation, we have $abc = -56$.<BR>\n" );
document.write( "We have<BR>\n" );
document.write( "\begin{align*} a+b+c &= \frac{x}{2} + 5y - 4z \\ ab+bc+ca &= \frac{x}{2}(5y) + 5y(-4z) + (-4z)\frac{x}{2} = \frac{5}{2}xy - 20yz - 2xz \\ abc &= \frac{x}{2}(5y)(-4z) = -10xyz\end{align*}<BR>\n" );
document.write( "The monic cubic polynomial is<BR>\n" );
document.write( "$$t^3 - (a+b+c)t^2 + (ab+bc+ca)t - abc = 0$$<BR>\n" );
document.write( "$$t^3 - \left(\frac{x}{2}+5y-4z\right)t^2 + \left(\frac{5}{2}xy-20yz-2xz\right)t - (-10xyz) = 0$$<BR>\n" );
document.write( "$$t^3 - \left(\frac{x}{2}+5y-4z\right)t^2 + \left(\frac{5}{2}xy-20yz-2xz\right)t + 10xyz = 0$$<BR>\n" );
document.write( "From the equations, we have<BR>\n" );
document.write( "\begin{align*} 15a + 2b - \frac{75}{4}c &= -150 \\ ab - \frac{105}{2}ac + bc &= 85 \\ abc &= -56\end{align*}<BR>\n" );
document.write( "We have $a+b+c = 2$, $ab+bc+ca = -21$, $abc = -56$.<BR>\n" );
document.write( "Thus the polynomial is $t^3 - 2t^2 - 21t + 56 = 0$.<BR>\n" );
document.write( "The roots are $t = -4, 2, 7$.<BR>\n" );
document.write( "Then $a, b, c$ can be $-4, 2, 7$ in any order.<BR>\n" );
document.write( "If $a=-4, b=2, c=7$, then $x=-8, y=2/5, z=-7/4$. $x+y+z = -8 + 2/5 - 7/4 = -8 - 1.35 = -9.35$.<BR>\n" );
document.write( "$x+y+z = -8 + 0.4 - 1.75 = -9.35$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{-9.35}$<BR>\n" );
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