Question

Доведіть тотожність (х/у²-1/х):(1/у-1/х)=х+у/у Даю 70балів до середи надо срочнооооооооооооо @@

Answer 1

Ответ:

$\displaystyle (\frac{x}{y^2}-\frac{1}{x}):(\frac{1}{y}-\frac{1}{x})=\frac{x+y}{y}$

Объяснение:

$\displaystyle (\frac{x}{y^2}-\frac{1}{x}):(\frac{1}{y}-\frac{1}{x})=(\frac{x\cdot x}{y^2\cdot x}-\frac{1\cdot y^2}{x\cdot y^2}):(\frac{1\cdot x}{y\cdot x}-\frac{1\cdot y}{x\cdot y})=\\\\\\(\frac{x^2}{xy^2}-\frac{y^2}{xy^2}):(\frac{x}{xy}-\frac{y}{xy})=\frac{x^2-y^2}{xy^2}:\frac{x-y}{xy}=\\\\\\\frac{(x-y)(x+y)}{xy^2}\cdot\frac{xy}{x-y}=\frac{x+y}{y}$

Answer 2

Solving the Equation

To solve the given equation and prove the identity (x/u² - 1/x) : (1/u - 1/x) = x + u/u, we can start by simplifying the left-hand side of the equation.

First, let's express the left-hand side of the equation in a common denominator:

(x/u² - 1/x) : (1/u - 1/x)

This can be rewritten as:

((x * x - u²) / (x * u²)) : ((x - u) / (x * u))

Simplifying further:

((x² - u²) / (x * u²)) : ((x - u) / (x * u))

Now, let's simplify the expression inside the brackets:

((x + u)(x - u) / (x * u²)) : ((x - u) / (x * u))

Canceling out the common factor (x - u):

((x + u) / u)

So, the left-hand side of the equation simplifies to (x + u) / u.

Verifying the Identity

Now, let's compare the simplified left-hand side with the right-hand side of the equation, which is x + u/u.

By simplifying the right-hand side, we get:

x + u/u

This simplifies to (x + u) / u.

Therefore, we have successfully proven the identity:

(x/u² - 1/x) : (1/u - 1/x) = x + u/u

This demonstrates that the given equation is indeed an identity.

Feel free to ask if you have further questions or need additional explanations!