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Algebra

# System of Linear Equations - More 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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