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Q. Solve for .Answer:
- Find Common Denominator: Find a common denominator for the fractions on the left side of the equation.The common denominator for and is . Multiply the first fraction by to get the common denominator.$(\frac{\(8\)}{\(7\)}) \cdot (\frac{\(6\)}{\(6\)}) + \frac{x - \(5\)}{\(42\)} = \(2\)x + \(3\)
- Simplify Left Side: Simplify the left side of the equation after finding the common denominator.\(\newline\)\((\frac{48}{42}) + \frac{(x - 5)}{42} = 2x + 3\)\(\newline\)Combine the fractions on the left side.\(\newline\)\((\frac{48 + x - 5}{42}) = 2x + 3\)
- Combine Fractions: Simplify the numerator of the combined fraction on the left side.\(\newline\)\((48 + x - 5) = 43 + x\)\(\newline\)So, \(\frac{43 + x}{42} = 2x + 3\)
- Simplify Numerator: Multiply both sides of the equation by \(42\) to eliminate the denominator.\(\newline\)\(42 \times \left(\frac{43 + x}{42}\right) = 42 \times (2x + 3)\)\(\newline\)\(43 + x = 84x + 126\)
- Multiply by \(42\): Subtract \(x\) from both sides to start isolating the variable \(x\).\(43 + x - x = 84x + 126 - x\)\(43 = 83x + 126\)
- Subtract \(x\): Subtract \(126\) from both sides to continue isolating \(x\).
\(43 - 126 = 83x + 126 - 126\)
\(-83 = 83x\) - Subtract \(126\): Divide both sides by \(83\) to solve for \(x\).\(\newline\)\(-\frac{83}{83} = \frac{83x}{83}\)\(\newline\)\(x = -1\)
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