Jos is 33 years younger than his father. The product of their ages is 658 . How old is Jos?

Let's denote Jos's age as \( J \) and his father's age as \( F \). Given: 1. Jos is 33 years younger than his father: \( J = F - 33 \) 2. The product of their ages is 658: \( J \times F = 658 \) We can solve these equations to find Jos's age. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}J=F-33\\JF=658\end{array}\right.\) - step1: Substitute the value of \(J:\) \(\left(F-33\right)F=658\) - step2: Simplify: \(F\left(F-33\right)=658\) - step3: Expand the expression: \(F^{2}-33F=658\) - step4: Move the expression to the left side: \(F^{2}-33F-658=0\) - step5: Factor the expression: \(\left(F-47\right)\left(F+14\right)=0\) - step6: Separate into possible cases: \(\begin{align}&F-47=0\\&F+14=0\end{align}\) - step7: Solve the equation: \(\begin{align}&F=47\\&F=-14\end{align}\) - step8: Calculate: \(F=47\cup F=-14\) - step9: Rearrange the terms: \(\left\{ \begin{array}{l}F=47\\J=F-33\end{array}\right.\cup \left\{ \begin{array}{l}F=-14\\J=F-33\end{array}\right.\) - step10: Calculate: \(\left\{ \begin{array}{l}F=47\\J=14\end{array}\right.\cup \left\{ \begin{array}{l}F=-14\\J=-47\end{array}\right.\) - step11: Calculate: \(\left\{ \begin{array}{l}F=-14\\J=-47\end{array}\right.\cup \left\{ \begin{array}{l}F=47\\J=14\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}F=-14\\J=-47\end{array}\right.\cup \left\{ \begin{array}{l}F=47\\J=14\end{array}\right.\) - step13: Rewrite: \(\left(F,J\right) = \left(-14,-47\right)\cup \left(F,J\right) = \left(47,14\right)\) The solutions to the system of equations are: 1. \( F = -14, J = -47 \) 2. \( F = 47, J = 14 \) Since age cannot be negative, we discard the solution where Jos's age is negative. Therefore, Jos is 14 years old.