Question
Solve for \( w \). \[ 2+\frac{5}{w-6}=\frac{3}{w-6} \] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". \[ \begin{array}{ll}\square\end{array} \]
Ask by Parry Kelly. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( w = 5 \).
Solution
To solve for w in the equation
2 + 5/(w − 6) = 3/(w − 6),
we first note that w cannot be 6 because that would make the denominator zero.
Now, multiply both sides of the equation by (w − 6):
2(w − 6) + 5 = 3.
Distribute on the left side:
2w − 12 + 5 = 3
2w − 7 = 3.
Add 7 to both sides:
2w = 10.
Finally, divide both sides by 2:
w = 5.
Since w = 5 does not make the denominator zero (w − 6 = −1), it is the valid solution.
The solution is 5.
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Simplify this solution Mind Expander
To solve the equation \[ 2+\frac{5}{w-6}=\frac{3}{w-6}, \] we first subtract \(\frac{3}{w-6}\) from both sides: \[ 2 + \frac{5}{w-6} - \frac{3}{w-6} = 0. \] This simplifies to: \[ 2 + \frac{2}{w-6} = 0. \] Now, we isolate the fraction: \[ \frac{2}{w-6} = -2. \] Next, we cross-multiply: \[ 2 = -2(w - 6). \] Distributing \(-2\) gives us: \[ 2 = -2w + 12. \] Now we rearrange it: \[ 2w = 12 - 2, \] \[ 2w = 10, \] \[ w = 5. \] Finally, we should check if \(w=5\) would make the original equation undefined by substituting back into the denominator \(w - 6\): \[ 5 - 6 = -1 \neq 0. \] Thus, the solution for \(w\) is: \[ \boxed{5}. \]
