if x,y,z satisfy xy=1 yz=2 and x=3 what is the value of x^2+y^2+z^2

Question

The question is for advanced math 

If x,y and z satisfy xy=1 yz=2 and xz=3 what is the value of x^2+y^2+z^2 express your answer as a common fraction.

Answer 1

You need to know the values of x, y, and z. It tells you that x=3, so you're a third of the way there already.

 

It tells you that xy=1. That means that x and y are reciprocals. The reciprocal of 3 is 1/3, so y=1/3.

 

Lastly, yz=2. You just figured out that y=1/3, so substitute to get 1/3 * z = 2.

 

Multiply each side of the equation by 3 to get z=6.

 

What is the value of x^2+y^2+z^2? Now that you know the values of all the variables, you can substitute to get:

 

3^2 + 1/3^2 + 6^2

 

Three squared is 9. 1/3 squared is 1/9. And, 6 squared is 36, so

 

9 + 1/9 + 36

 

That equals 45 1/9. If you want to write that as an improper fraction, of course it is 406/9.

Answer 2

Substitute given value  x=3 into the equation  xy=1.

 

y = 1 / x

y = 1 / 3

 

 

Substitute the value of y into the equation yz=2.

 

z = 2 / y

z = 2 / (1 / 3)

z = 2 * 3

z = 6

 

 

Now substitute these values into the expression.

 

x² + y² + z² =

 

3² + (1 / 3)² + 6² =

 

9 + (1 / 9) + 36

 

 

Using the LCD of 9.

 

(81 / 9) + (1 / 9) + (324 / 9)

 

 

Now complete the calculation.