Questions: Solve for (w). [ -frac12 w-2=frac54 w-frac73 ] Simplify your answer as much as possible. [ w= ]

Solve for (w). [ -frac12 w-2=frac54 w-frac73 ] Simplify your answer as much as possible. [ w= ]
Transcript text: Solve for $w$. \[ -\frac{1}{2} w-2=\frac{5}{4} w-\frac{7}{3} \] Simplify your answer as much as possible. \[ w= \]
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Solution

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Solution Steps

To solve the linear equation for \( w \), we need to isolate \( w \) on one side of the equation. This involves combining like terms and performing basic arithmetic operations to simplify the equation.

Step 1: Combine Like Terms

First, we start with the given equation: \[ -\frac{1}{2} w - 2 = \frac{5}{4} w - \frac{7}{3} \]

Step 2: Eliminate Fractions

To eliminate the fractions, we find a common denominator. The common denominator for 2, 4, and 3 is 12. Multiply every term by 12: \[ 12 \left( -\frac{1}{2} w \right) - 12 \cdot 2 = 12 \left( \frac{5}{4} w \right) - 12 \left( \frac{7}{3} \right) \] \[ -6w - 24 = 15w - 28 \]

Step 3: Isolate the Variable

Next, we move all terms involving \( w \) to one side and constant terms to the other side: \[ -6w - 15w = -28 + 24 \] \[ -21w = -4 \]

Step 4: Solve for \( w \)

Divide both sides by -21 to solve for \( w \): \[ w = \frac{-4}{-21} = \frac{4}{21} \approx 0.1905 \]

Final Answer

\[ \boxed{w = 0.1905} \]

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