**Solution:**

Given equations are 3x-4y=10 and xy=10.

We need to find the value of 9x^2+17y^2.

Let's solve the given equations to find the values of x and y.

Multiplying both sides of the equation xy=10 by 4, we get

4xy = 40

Adding this equation to 3x-4y=10, we get

3x + 4xy - 4y = 50

Substituting xy=10, we get

3x + 40 - 4y = 50

3x - 4y = 10

This is the same as the first equation given to us.

So, we have two equations with two variables:

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

xy = 10 ...(2)

**Finding values of x and y**

We can solve for one variable in terms of the other, and substitute in the other equation.

From equation (2), we get

y = 10/x

Substituting this in equation (1), we get

3x - 4(10/x) = 10

Multiplying both sides by x, we get

3x^2 - 40 = 10x

Bringing all the terms to one side, we get

3x^2 - 10x - 40 = 0

Solving this quadratic equation, we get

x = 4 or -10/3

Substituting these values in equation (2), we get

y = 10/4 = 2.5 or y = -10/(-10/3) = 3

So, we have two solutions: (x,y) = (4,2.5) or (-10/3, 3)

**Finding the value of 9x^2+17y^2**

Let's substitute each of these solutions in the expression 9x^2+17y^2 to find the value.

For (x,y) = (4,2.5), we get

9x^2+17y^2 = 9(4)^2 + 17(2.5)^2 = 144 + 106.25 = 250.25

For (x,y) = (-10/3,3), we get

9x^2+17y^2 = 9(-10/3)^2 + 17(3)^2 = 300 + 459 = 759

Therefore, the value of 9x^2+17y^2 is either 250.25 or 759, depending on which solution we choose.

Hence, the solution.