1. if x/a + y/b = a² + b² and x/a² + y/b² = a +b, then find x.
2. Solve the equations: 4(2^x-1) + 9(3^y-1) = 17 and 3(2^x) – 2(3^y) = 6

Hello student,
Please find the answer to your question below
1)Given Equations are
x/a^2 + y/b^2 = 2..........(1)
x/a + y/b = a + b...........(2)
Simplifying equations (1) and (2) by taking lcm we get
b²x+a²y=2a²b²..........(4)
bx+ay=a²b+ab²........(3)
Multiplying equation 3 with b we get
b²x+aby=a²b²+ab³.........(5)
b²x+a²y=2a²b²............(6)
subtracting 5 from 6 we get
(ab-a²)y=ab³-a²b²
a(b-a)y=ab²(b-a)
After cancelling a(b-a) on both sides we get y=b²
substituting y=b²in x/a + y/b = a + b we get x=a²
2)Let4(2^x-1) + 9(3^y-1) = 17......(1)and 3(2^x) – 2(3^y) = 6.......(2)
(2^x-1+2)+(3^y-1+2)=17
(2^x+1)+(3^y+1)=17
2(2^x)+3(3^y)=17........(3) and from 2 we have 3(2^x)-2(3^y)=6
Multiplying eqn 3 by 3 and eqn 2 by 2 we get
6(2^x)+9(3^y)=51
6(2^x)-4(3^y)=12
subtracting above 2 equations we get
13(3^y)=39 so 3^y=3¹So y=1
Substituting y=1 in 3(2^x)-2(3)=6
3(2^x)=12 So 2^x=4=2² so x=2