Question Solve the equation $6+\frac{6}{w-3}=\frac{5}{w+1}$ for the unknown variable $w$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $6+\frac{6}{w-3}=\frac{5}{w+1}$ .
2. We assume that $w \neq 3$ and $w \neq -1$ to avoid division by zero in the denominators.

STEP 2

To solve the equation, we want to eliminate the fractions by finding a common denominator. The common denominator for the fractions $\frac{6}{w-3}$ and $\frac{5}{w+1}$ is $(w-3)(w+1)$ .

STEP 3

Multiply each term in the equation by the common denominator $(w-3)(w+1)$ to clear the fractions.
$(w-3)(w+1) \cdot 6 + (w-3)(w+1) \cdot \frac{6}{w-3} = (w-3)(w+1) \cdot \frac{5}{w+1}$

STEP 4

Distribute the common denominator across the terms on both sides of the equation.
$6(w-3)(w+1) + 6(w+1) = 5(w-3)$

STEP 5

Simplify each term. For the first term, we distribute $6$ into $(w-3)(w+1)$ , and for the second term, we cancel out $(w-3)$ in the numerator and denominator. For the third term, we cancel out $(w+1)$ in the numerator and denominator.
$6(w^2 - 3w + w - 3) + 6(w+1) = 5(w-3)$

STEP 6

Combine like terms in the first expression.
$6(w^2 - 2w - 3) + 6(w+1) = 5(w-3)$

STEP 7

Distribute $6$ into the quadratic expression and $6$ into $(w+1)$ .
$6w^2 - 12w - 18 + 6w + 6 = 5w - 15$

STEP 8

Combine like terms on the left side of the equation.
$6w^2 - 6w - 12 = 5w - 15$

STEP 9

Move all terms to one side to set the equation to zero and combine like terms.
$6w^2 - 6w - 5w - 12 + 15 = 0$
$6w^2 - 11w + 3 = 0$

STEP 10

Now we have a quadratic equation. We can attempt to factor it, or we can use the quadratic formula to find the values of $w$ . Let's try to factor it first.

STEP 11

Look for two numbers that multiply to $6 \times 3 = 18$ and add up to $-11$ .

STEP 12

The numbers that satisfy these conditions are $-9$ and $-2$ because $-9 \times -2 = 18$ and $-9 + -2 = -11$ .

STEP 13

Rewrite the quadratic equation using these two numbers to split the middle term.
$6w^2 - 9w - 2w + 3 = 0$

STEP 14

Factor by grouping. Group the first two terms and the last two terms separately.
$3w(2w - 3) - 1(2w - 3) = 0$

STEP 15

Factor out the common binomial factor $(2w - 3)$ .
$(3w - 1)(2w - 3) = 0$

STEP 16

Set each factor equal to zero to solve for $w$ .
$3w - 1 = 0 \quad \text{or} \quad 2w - 3 = 0$

STEP 17

Solve each equation for $w$ .
$3w = 1 \quad \text{or} \quad 2w = 3$

STEP 18

Divide by the coefficients to isolate $w$ .
$w = \frac{1}{3} \quad \text{or} \quad w = \frac{3}{2}$
The solutions to the equation are $w = \frac{1}{3}$ and $w = \frac{3}{2}$ .

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