Matej Anton age

Matthew and Anthony are 44 years old together. Matthew is twice as old as Anthony was when Matthew was half as old as Anthony will be when Anthony is 3 times older than Matthew was when Matthew was 3 times as old as Anthony.

Final Answer:

M =  34.5714
A =  9.4286

Step-by-step explanation:

$$\begin{gathered} M+A=44 \\ M=2\cdot (A-x) \\ (M-x)=0.5\cdot (A+y) \\ (A+y)=3\cdot (M-z) \\ (M-z)=3\cdot A \\ A+M=44 \\ 2A-M-2x=0 \\ 0.5A-M+x+0.5y=0 \\ A-3M+y+3z=0 \\ 3A-M+z=0 \\ Pivot:Row1\leftrightarrow Row5 \\ Row2-\frac{2}{3}\cdot Row1\to Row2 \\ -0.33333M-2x-0.66667z=0 \\ Row3-\frac{0.5}{3}\cdot Row1\to Row3 \\ -0.83333M+x+0.5y-0.16667z=0 \\ Row4-\frac{1}{3}\cdot Row1\to Row4 \\ -2.66667M+y+2.66667z=0 \\ Row5-\frac{1}{3}\cdot Row1\to Row5 \\ 1.33333M-0.33333z=44 \\ Pivot:Row2\leftrightarrow Row4 \\ Row3-\frac{-0.83333333}{-2.66666667}\cdot Row2\to Row3 \\ x+0.1875y-z=0 \\ Row4-\frac{-0.33333333}{-2.66666667}\cdot Row2\to Row4 \\ -2x-0.125y-z=0 \\ Row5-\frac{1.33333333}{-2.66666667}\cdot Row2\to Row5 \\ 0.5y+z=44 \\ Pivot:Row3\leftrightarrow Row4 \\ Row4-\frac{1}{-2}\cdot Row3\to Row4 \\ 0.125y-1.5z=0 \\ Pivot:Row4\leftrightarrow Row5 \\ Row5-\frac{0.125}{0.5}\cdot Row4\to Row5 \\ -1.75z=-11 \\ z=\frac{-11}{-1.75}=6.28571429 \\ y=\frac{44-z}{0.5}=\frac{44-6.28571429}{0.5}=75.42857143 \\ x=\frac{0+0.125y+z}{-2}=\frac{0+0.125\cdot 75.42857143+6.28571429}{-2}=-7.85714286 \\ M=\frac{0-y-2.6666666666667z}{-2.66666667}=\frac{0-75.42857143-2.66666667\cdot 6.28571429}{-2.66666667}=34.57142857 \\ A=\frac{0+M-z}{3}=\frac{0+34.57142857-6.28571429}{3}=9.42857143 \\ A=\frac{66}{7}\doteq 9.428571 \\ M=\frac{242}{7}\doteq 34.571429 \\ x=\frac{-55}{7}\doteq -7.857143 \\ y=\frac{528}{7}\doteq 75.428571 \\ z=\frac{44}{7}\doteq 6.285714 \end{gathered}$$

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