Problems 16-23
16 (a) If $X$ is twice as old as $Y$ and their ages add to $33$, then
$$ X + Y = 2Y + Y = 33. $$The equation can be expressed as linear system
$$ \mathbf{Ab} = \begin{bmatrix} 2 & 1 \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} Y \\ Y \end{bmatrix} = \begin{bmatrix} 3Y \\ 0 \end{bmatrix} = \begin{bmatrix} 33 \\ 0 \end{bmatrix} = \mathbf{b}. $$The first component of $\mathbf{Ab}$ is $3Y$ $=$ $33$, so $Y$ $=$ $11$. Substituting this back to the original equation means that
$$ X + 11 = 33, $$so $X$ $=$ $22$.
(b)
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