Questions: Solve for (v). [ frac6v-6=fracvv-6+3 ] [ v= ]

$Solve for (v). [ frac6v-6=fracvv-6+3 ] [ v= ]$

Transcript text: Solve for $v$. \[ \frac{6}{v-6}=\frac{v}{v-6}+3 \] \[ v= \]

Solution

Solution Steps

Step 1: Combine the Fractions

We start with the equation: \[ \frac{6}{v-6} = \frac{v}{v-6} + 3 \] Since both sides have the same denominator $ v - 6 $, we can eliminate the denominators by multiplying through by $ v - 6 $ (assuming $ v \neq 6 $): \[ 6 = v + 3(v - 6) \]

Step 2: Simplify the Equation

Expanding the right-hand side gives: \[ 6 = v + 3v - 18 \] Combining like terms results in: \[ 6 = 4v - 18 \]

Step 3: Solve for $ v $

Next, we isolate $ v $ by adding 18 to both sides: \[ 24 = 4v \] Dividing both sides by 4 yields: \[ v = 6 \]

Step 4: Check for Validity

However, substituting $ v = 6 $ back into the original equation leads to a division by zero in the denominators, which is undefined. Therefore, $ v = 6 $ is not a valid solution.

Final Answer

Since there are no valid solutions, we conclude: \[ \boxed{\text{No solution}} \]

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