Originally Posted by
Soroban Hello, magentarita!
I assume your mother's name is Anna . . .
Let $\displaystyle M$ = my age (now).
Let $\displaystyle A$ = Anna's age (now).
Seven years from now, both of us are 7 years older.
. . $\displaystyle \begin{array}{ccc}\text{My age} &=& M + 7 \\ \text{Her age} &=& A + 7 \end{array}$
. . $\displaystyle \underbrace{\text{My age}}_{M+7}\: \underbrace{\text{will be}}_{=}\: \underbrace{\text{half her age}}_{\frac{1}{2}(A+7)}$
We have: .$\displaystyle M + 7 \:=\:\tfrac{1}{2}(A+7) \quad\Rightarrow\quad A - 2M \:=\:7\;\;{\color{blue}[1]}$
Thirteen years ago, both of us were 13 years younger.
. . $\displaystyle \begin{array}{ccc}\text{My age} &=& M-13 \\ \text{Her age} &=& A - 13 \end{array}$
. . $\displaystyle \underbrace{\text{Her age}}_{A-13}\: \underbrace{\text{was}}_{=}\: \underbrace{\text{6 times my age}}_{6(M-13)} $
We have: .$\displaystyle A - 13 \:=\:6(M-13) \quad\Rightarrow\quad A - 6M \:=\:-65\;\;{\color{blue}[2]} $
Subtract [2] from [1]: .$\displaystyle 4M \:=\:72 \quad\Rightarrow\quad M \:=\:18$
Substitute into [1]: .$\displaystyle A -2(18) \:=\:7 \quad\Rightarrow\quad\boxed{ A \:=\:43}$
Therefore, Anna is 43 years old.