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Systems of Linear Equations

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Many Variables

Consider a system of linear equations \[\begin{align} x+y &= 3, \\ y+z &= 11, \\ z+x &= 20. \end{align}\] What is the value of \(3x+y+5z\)?

Let \(x=a\), \(y=b\) and \(z=c\) be the values of \(x\), \(y\) and \(z\), respectively, that satisfy the system of linear equations \[ \begin{align} x-y&=3 \\ y+z&=25 \\ z-x&=8. \end{align}\] What is the value of \(a+b+c\)?

Let \(x=a\), \(y=b\) and \(z=c\) be the values of \(x\), \(y\) and \(z\), respectively, that satisfy the system of linear equations \[ \begin{align} x+2y&= 24 \\ 2y+3z&= 58\\ x+3z&= 38. \end{align}\] What is the value of \(a+b+c\)?

Let \(x=a\), \(y=b\) and \(z=c\) be the values of \(x\), \(y\) and \(z\), respectively, that satisfy the system of linear equations \[ \frac{3-x}{2}= \frac{y+1}{3}=\frac{2z-1}{5}, \, 3x+2y+z=65. \] What is the value of \(a+b+c\)?

Let \(x=A\), \(y=B\) and \(z=C\) be the solutions of the simultaneous equations \[\begin{align} x+y+z &= 270 \\ x-y+z &= 90 \\ x+y-z &=0. \end{align}\] What is the value of \(\frac{A+B}{C}\)?

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