Respuesta :
To answer this question, we can proceed as follows:
1. Let x be Amy's age.
2. Let y be Jonny's age.
Six years ago, we have that the ages were:
A ---> x - 6
J ---> y - 6
And we have that six years ago, Amy was twice as old Jonny. Then, we have:
[tex](x-6)=2(y-6)[/tex]Currently, Amy is 7 years older than Jonny:
[tex]x=y+7[/tex]Now we have the next equations:
[tex](x-6)=2(y-6)[/tex]And
[tex]x=y+7[/tex]Expanding the first equation:
[tex](x-6)=2(y-6)\Rightarrow x-6=2y-12\Rightarrow x-2y=-12+6[/tex]Then, we have:
[tex]x-2y=-6[/tex]And the other equation can be rewritten as:
[tex]x=y+7\Rightarrow x-y=7[/tex]Then, we have the following system of equations:
[tex]\begin{cases}x-2y=-6 \\ x-y=7\end{cases}[/tex]Now, to solve this system, we can multiply the first equation by -1, and then add both resulting equations as follows:
[tex]\frac{\begin{cases}-1(x-2y=-6)=-x+2y=6 \\ x-y=7\Rightarrow x-y=7\end{cases}}{}[/tex][tex]\frac{\begin{cases}-x+2y=6 \\ x-y=7\end{cases}}{y=13}[/tex]Then, we have that Jonny's age is 13 years old. Therefore, Amy's age is:
[tex]x=y+7\Rightarrow x=13+7\Rightarrow x=20[/tex]Then, Amy is 20 years old, and Jonny is 13 years old.
We can check the result using the age of both of them six years ago:
Amy ---> 20 - 6 = 14
Jonny ---> 13 - 6 = 7
As we can see, 6 years ago Amy was twice as old as Jonny.
In summary, we have that Amy's age is 20 years old and Jonny is 13 years old.
