Question

1. 2x-8/x-3 + x-3/5x+5 =3 1/2 2. 1/y+3 = y+9/4y+20

Answer 1

Ответ:

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Answer 2

Let's solve these equations step by step:

Equation 1: $\frac{2x - 8}{x - 3} + \frac{x - 3}{5x + 5} = \frac{7}{2}$

First, find a common denominator for the fractions on the left-hand side. The common denominator will be $(x - 3)(5x + 5)$.

Now, rewrite each fraction with the common denominator:

$\frac{(2x - 8)(5x + 5)}{(x - 3)(5x + 5)} + \frac{(x - 3)(x - 3)}{(x - 3)(5x + 5)} = \frac{7}{2}$

Now that the fractions have a common denominator, you can combine them:

$(2x - 8)(5x + 5) + (x - 3)(x - 3) = \frac{7}{2}(x - 3)(5x + 5)$

Expand and simplify:

$10x^2 - 36x + 25x - 40 + x^2 - 6x + 9 = \frac{7}{2}(5x^2 - 15x - 15x - 45)$

Combine like terms:

$11x^2 - 11x - 31 = \frac{7}{2}(5x^2 - 30x - 45)$

Now, multiply both sides by 2 to get rid of the fraction:

$22x^2 - 22x - 62 = 7(5x^2 - 30x - 45)$

Expand further:

$22x^2 - 22x - 62 = 35x^2 - 210x - 315$

Subtract $35x^2$ and $210x$ from both sides:

$22x^2 - 35x^2 - 22x + 210x - 62 + 315 = 0$

Combine like terms:

$-13x^2 + 188x + 253 = 0$

Now, you have a quadratic equation. To solve for $x$, you can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = -13$, $b = 188$, and $c = 253$. Plug these values into the quadratic formula to find the solutions for $x$. You may need a calculator for the numerical values.

Equation 2: $\frac{1}{y + 3} = \frac{y + 9}{4y + 20}$

Cross-multiply to get rid of the fractions:

$(y + 3)(y + 9) = 1(4y + 20)$

Expand and simplify:

$y^2 + 12y + 27 = 4y + 20$

Subtract $4y + 20$ from both sides:

$y^2 + 12y + 27 - 4y - 20 = 0$

Combine like terms:

$y^2 + 8y + 7 = 0$

This is a quadratic equation in $y$. You can solve for $y$ using the quadratic formula:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$