For this case we propose a system of equations:
x: Let the variable representing the age of the first child of the Smiths
y: Let the variable representing the age of the second child of the Smiths
According to the data of the statement we have to:
[tex]x + y = 23\\x * y = 132[/tex]
From the first equation we have to:
[tex]x = 23-y[/tex]
We substitute in the second equation:
[tex](23-y) * y = 132\\23y-y ^ 2 = 132\\y ^ 2-23y + 132 = 0[/tex]
We find the solutions by factoring:
We look for two numbers that, when multiplied, result in 132 and when added, result in 23. These numbers are 11 and 12.
Thus, we have that the factorized equation is:
[tex](y-11) (y-12) = 0[/tex]
Thus, the solutions are:[tex]y_ {1} = 11\\y_ {2} = 12[/tex]
So, we can take any of the solutions:
With [tex]y = 11[/tex]
Then[tex]x = 23-11 = 12[/tex]
Therefore, the ages of the children are 11 and 12 respectively.
Answer:
The ages of the children are 11 and 12 respectively.