You can divide the equations, to get
$$\begin{align}
\frac1x + \frac1y &= \frac56 = \frac{35}{42}\\
\frac1y+\frac1z &= \frac{10}{21} = \frac{20}{42}\\
\frac1x+\frac1z &= \frac{9}{14} = \frac{27}{42}
\end{align}$$
Subtracting the second equation from the first yields $$\frac1x - \frac1z = \frac{15}{42}$$
and adding that to the last
$$\frac{2}{x} = \frac{42}{42} = 1 \Rightarrow x = 2.$$
Subtracting from the last yields
$$\frac{2}{z} = \frac{12}{42} \Rightarrow z = 7.$$
Inserting $x = 2$ for example into the first equation yields $y = 3$.
- $P$ Pa
- $M$ Ma
- $B$ Brother
- $Y$ You
$$P + M + B + Y = 83$$
$$6P = 7M$$
$$M = 3Y$$
Combine and write everything in terms of $B$ and $Y$.
$$21/6 Y + 3Y + B + Y = 83$$
$$45 Y + 6B = 498$$
Sum of even numbers is even, so $Y$ must be even, call it $Y=2n$:
$$90n + 6Y = 498$$
$$15n + Y = 83$$
$$\begin{bmatrix} n \\ B \end{bmatrix} = \begin{bmatrix} n \\ 83 - 15n \end{bmatrix}$$
$$\begin{bmatrix} Y \\ B \end{bmatrix} = \begin{bmatrix} 2n \\ 83 - 15n \end{bmatrix}$$
Insert back in the parents:
$$\begin{bmatrix} P \\ M \\ Y \\ B \end{bmatrix} = \begin{bmatrix} 7n \\ 6n \\ 2n \\ 83 - 15n \end{bmatrix}$$
I think we can assume $B \ge 0$, so $83 - 15n \ge 0$, so $n \le 5$.
Hopefully $B < M$, so $83 - 15n < 6n$, so $n \ge 4$.
So your choices are $n=4$ or $n=5$:
$$\begin{bmatrix} P \\ M \\ Y \\ B \end{bmatrix} = \begin{bmatrix} 28 \\ 24 \\ 8 \\ 23 \end{bmatrix} \text{ or } \begin{bmatrix} 35 \\ 30 \\ 10 \\ 8 \end{bmatrix}$$
Assuming you are humans and your brother isn't adopted, it's probably the second one.
Best Answer
Denote $x$ and $y$ the two numbers $x<y$ so we have
$$y-x=3\quad;\quad 4x=5y$$ Can you take it from here?