The ages of mother and daughter

Statement

My mother’s age $4$ years ago, added to twice my current age, gives $71$. My age $4$ years ago, added to three times my mother’s current age, gives $146$. How old is my mother now?

Solution

Let $m$ be the mother’s current age and $f$ my current age. $\begin{cases} (m-4) + 2f = 71\\ (f-4) + 3m = 146 \end{cases} \implies \begin{cases} m + 2f = 75\\ 3m + f = 150 \end{cases}$ From the first, $m = 75 - 2f$; substitute into the second: $3(75 - 2f) + f = 150 \implies 225 - 6f + f = 150 \implies -5f = -75 \implies f = 15.$ Then $m = 75 - 2\cdot 15 = 45$. Check: $(45-4) + 2\cdot15 = 41+30 = 71$ and $(15-4) + 3\cdot45 = 11+135 = 146$. Correct. $\boxed{\text{mother } 45 \text{ years}, \quad \text{daughter } 15 \text{ years}}$