The problem provides the information:
3 x + 5 y = 11
and x y = 2
Asking to find the value of 9 x^2 + 25 y^2
so we proceed to recall what the square of a binimial of the following form is:
[tex](3x+5y)^2=(3x)^2+2(3x\cdot5y)+(5y)^2=9x^2+30xy+25y^2_{}[/tex]So, we see that the square of the quantity: (3 x + 5 y) which from the information given equals 11^2 = 121, must equal the expression:
9 x^2 + 25 y^2 + 30 x y.
then we write the following equation making both parts equal:
[tex]\begin{gathered} 11^2=9x^2+25y^2+30xy \\ 121=9x^2+25y^2+30xy \\ 121=9x^2+25y^2+30(2) \\ 121-60=9x^2+25y^2 \\ 61=9x^2+25y^2 \end{gathered}[/tex]Therefore the value of 9 x^2 + 25 y^2 = 61
