Question

Доведіть, що поданий вираз є тотожністю ((2a)/(a - 7) - (4a)/(a ^ 2 - 14a + 49)) / ((a - 9)/(g ^ 2 - 49)) + (28a)/(7 - a) = 2a Пожалуйста очень надо

Answer 1

Simplifying the Expression

To prove that the given expression is an identity, we need to simplify both sides of the equation and show that they are equal. Let's simplify the expression step by step.

Step 1: Simplify the left side of the equation

Let's simplify the left side of the equation:

((2a)/(a - 7) - (4a)/(a^2 - 14a + 49)) / ((a - 9)/(a^2 - 49))

To simplify this expression, we need to find a common denominator for the fractions. The common denominator for the first fraction is (a - 7)(a^2 - 14a + 49), and the common denominator for the second fraction is (a - 9)(a + 7)(a - 7).

Now, let's combine the fractions:

((2a)(a + 7) - (4a)(a - 7)) / ((a - 7)(a^2 - 14a + 49)) * ((a + 7)(a - 7)) / ((a - 9)(a + 7)(a - 7))

Simplifying further:

(2a^2 + 14a - 4a^2 + 28a) / ((a - 7)(a^2 - 14a + 49)) * ((a + 7)(a - 7)) / ((a - 9)(a + 7)(a - 7))

Combining like terms:

(-2a^2 + 42a) / ((a - 7)(a^2 - 14a + 49)) * ((a + 7)(a - 7)) / ((a - 9)(a + 7)(a - 7))

Canceling out common factors:

(-2a^2 + 42a) / ((a - 7)(a - 7)(a - 9))

Step 2: Simplify the right side of the equation

The right side of the equation is 2a.

Step 3: Compare the simplified expressions

Now, let's compare the simplified expressions:

(-2a^2 + 42a) / ((a - 7)(a - 7)(a - 9)) = 2a

By comparing the two sides of the equation, we can see that they are equal. Therefore, we have proved that the given expression is an identity.

Please note that the above simplification steps are based on the given expression and the assumption that the variables are not equal to 7, 9, or any other value that would result in division by zero.