Question

solve for w.
\\(\frac{9w + 2}{7} = \frac{2w - 7}{5} + 7\\)
simplify your answer as much as possible.
\\(w = \boxed{}\\)

Explanation:

Step1: Eliminate denominators (LCM=35)

Multiply each term by 35:
$$35 \times \frac{9w+2}{7} = 35 \times \frac{2w-7}{5} + 35 \times 7$$
Simplify to get:
$$5(9w+2) = 7(2w-7) + 245$$

Step2: Expand all parentheses

$$45w + 10 = 14w - 49 + 245$$

Step3: Simplify right-hand side

Combine constant terms:
$$45w + 10 = 14w + 196$$

Step4: Isolate w terms

Subtract $14w$ from both sides:
$$45w - 14w + 10 = 196$$
$$31w + 10 = 196$$

Step5: Isolate the variable w

Subtract 10 from both sides:
$$31w = 196 - 10$$
$$31w = 186$$

Step6: Solve for w

Divide by 31:
$$w = \frac{186}{31}$$
$$w = 6$$

Answer:

$w = 6$