Question

solve.
\\(\frac{w}{w - 5} - \frac{4w}{4w - 5} = \frac{w - 1}{4w^2 - 25w + 25}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice
\\(\bigcirc\\) a. \\(w = \square\\)
\\(\quad\\) (type an integer or a simplified fraction use a comma to separate answers as needed )
\\(\bigcirc\\) b. the solution is all real numbers
\\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Factor the denominator on the right

First, factor \(4w^2 - 25w + 25\). We need two numbers that multiply to \(4\times25 = 100\) and add to \(-25\). Those numbers are \(-20\) and \(-5\). So,
\[

$$\begin{align*} 4w^2 - 25w + 25&=4w^2 - 20w - 5w + 25\\ &=4w(w - 5) - 5(w - 5)\\ &=(4w - 5)(w - 5) \end{align*}$$

\]
The equation becomes \(\frac{w}{w - 5}-\frac{4w}{4w - 5}=\frac{w - 1}{(4w - 5)(w - 5)}\).

Step2: Multiply through by the LCD

The least common denominator (LCD) of the fractions is \((4w - 5)(w - 5)\). Multiply each term by the LCD to eliminate the fractions:
\[

$$\begin{align*} w(4w - 5)-4w(w - 5)&=w - 1\\ 4w^2 - 5w - 4w^2 + 20w&=w - 1 \end{align*}$$

\]

Step3: Simplify and solve for \(w\)

Simplify the left - hand side:
\[

$$\begin{align*} (4w^2-4w^2)+(-5w + 20w)&=w - 1\\ 15w&=w - 1 \end{align*}$$

\]
Subtract \(w\) from both sides:
\[

$$\begin{align*} 15w-w&=w - 1-w\\ 14w&=-1 \end{align*}$$

\]
Divide both sides by 14: \(w =-\frac{1}{14}\)

Step4: Check for excluded values

We need to check if \(w =-\frac{1}{14}\) makes any of the original denominators zero.

• For \(w-5\), when \(w =-\frac{1}{14}\), \(w - 5=-\frac{1}{14}-5=-\frac{1 + 70}{14}=-\frac{71}{14}

eq0\)

• For \(4w - 5\), when \(w=-\frac{1}{14}\), \(4w-5=4\times(-\frac{1}{14})-5=-\frac{2}{7}-5=-\frac{2 + 35}{7}=-\frac{37}{7}

eq0\)

Answer:

A. \(w =-\frac{1}{14}\)