Hard math problem please help!?

Question

The real numbers x, y and z satisfy x − y = y − z = 10. What is x^2 − 2y^2 + z^2?

Can you please explain how to do this problem in detail, like what formulas to use, step by step, be my tutor for this problem =D ty.

Answer 1

Here's how you do it:

first take (x-y) and square it:

x^2 - 2*xy + y^2

then take (y-z) and square it:

y^2 - 2*yz + z^2

then you add (x-y)^2 and (y-z)^2     --->

x^2 + 2*(y^2) + z^2 - 2*xy - 2*yz

then you factor the last two terms:

= x^2 + 2*(y^2) + z^2 - 2y*(x+z)

but if you take the original equation, you know that:

x-y = y-z  ----->  x+z = 2y

you subsitute this back into the other equation and:

= x^2 + 2*(y^2) + z^2 - 2y*(2y) = x^2 - 2y^2 + z^2

this is the equation you want.

You know the value and it is 10^2 + 10^2 = 200

then answer is 200  

Answer 2

x − y = y − z = 10 (1)

(1) <=>x+z=2y<=>x^2+z^2+2xz=4y^2 <=>x^2+z^2-2y^2=2y^2-2xz (*)

(1)<=>z=y-10(**)

(1)<=>x=10+y(***)

(*),(**)&(***)==>x^2 − 2y^2 + z^2=2y^2-2(10+y)(y-10)

<=>x^2 − 2y^2 + z^2=2y^2-2(y^2-100)=2(y^2-y^2+100)=200

Answer 3

Ok, we'll go step by step

x − y = 10

y − z = 10

(x - y) + (y - z) = (x - z)

10 + 10 = (x - z)

20 = (x - z)

x² − 2y² + z²

x² − y² + z² - y²

(x – y)(x + y) + (z – y)(z + y)

10(x + y) – 10(z + y)

10x + 10y – 10z – 10y

10(x – z)

Use (x – z) = 10 from above

10*20 = 200

Answer: 200

Answer 4

x-y=y-z=10

So, y=x-10

      Replacing it in 2nd Equation

x-10-z=10 ... So.. x-z=20 Therfore, x=20+z       ......1

y=20+z-10 = 10+z                                           ......2

z=z

x^2-2y^2+z^2 = (20+z)^2-2(10+z)^2+z^2   =  200

Henceforth the answer is 200.

Don't 4get to mark my answer as the best ..hahaha

Bye

tc

Answer 5

x^2 − 2y^2 + z^2

= x^2 - y^2 - (y^2 - z^2)

= (x - y)(x + y) - (y - z)(y + z)

= 10(x + y) - 10(y + z)

= 10(x + y - y - z)

= 10(x - z)

= 10*20 (because x - z = x - y + y - z = 10 + 10 = 20)

= 200.

Answer 6

Let's break down the second part as follows:

x² - 2y² + z²

= x² - y² - y² + z²

Then again as:

(x² - y²) - (y² - z²)

Rewrite the difference of squares:

(x - y)(x + y) - (y - z)(y + z)

Substitute in the values of 10:

10(x + y) - 10(y + z)

Distribute the 10s through the parentheses:

10x + 10y - 10y - 10z

Cancel:

10x - 10z

10(x - z)

Now solve the other equations for x and z in terms of y:

x = 10 + y

-z = 10 + y

10(10 + y + 10 + y)

10(20 + 2y)

20(10 + y)

= 20x