3x+4y=16
Squaring both sides
=>(3x+4y)2=256
=>9x2+16y2+2(3x)(4y)=256
=>9x2+16y2+24xy=256
=>9x2+16y2=256−24×4
=>9x2+16y2=256−96
=>9x2+16y2=160

To find the value of 9x^2 - 16y^2, we first need to solve the given system of equations:

3x - 4y = 10 ........(1)
xy = -1 ...........(2)

Solving the System of Equations:
Let's solve the given system of equations using the substitution method.

From equation (2), we can express y in terms of x as y = -1/x.

Substituting this value of y in equation (1), we get:

3x - 4(-1/x) = 10

Simplifying this equation, we get:

3x + 4/x = 10

Multiplying both sides of the equation by x, we eliminate the denominator:

3x^2 + 4 = 10x

Rearranging the equation, we get:

3x^2 - 10x + 4 = 0

Now, we can solve this quadratic equation using factoring or the quadratic formula. Let's use factoring to find the values of x.

Factoring the quadratic equation, we get:

(3x - 2)(x - 2) = 0

Setting each factor equal to zero, we have:

3x - 2 = 0 or x - 2 = 0

Solving for x, we get:

3x = 2 or x = 2

x = 2/3 or x = 2

Now that we have the values of x, we can substitute them back into equation (2) to find the corresponding values of y.

For x = 2/3:

xy = -1
(2/3)y = -1
y = -3/2

For x = 2:

xy = -1
2y = -1
y = -1/2

Therefore, the solutions to the system of equations are:
(x, y) = (2/3, -3/2) and (2, -1/2)

Finding the Value of 9x^2 - 16y^2:
Now that we have the values of x and y, we can substitute them back into the expression 9x^2 - 16y^2 to find its value.

For (x, y) = (2/3, -3/2):

9x^2 - 16y^2 = 9(2/3)^2 - 16(-3/2)^2
= 9(4/9) - 16(9/4)
= 4 - 36
= -32

For (x, y) = (2, -1/2):

9x^2 - 16y^2 = 9(2)^2 - 16(-1/2)^2
= 9(4) - 16(1/4)
= 36 - 4
= 32

Therefore, the value of 9x^2 - 16y^2 is -32 for (x, y) = (2/3, -3/2), and 32 for (x, y) = (2, -1/2).