Error Analysis 122D8F

1. **State the problem:** We are given the equation $$\frac{2}{3} x + 7 \frac{5}{6} = \frac{1}{2} x + 9$$ and a series of steps attempting to solve for $$x$$. We need to identify errors in the solution process. 2. **Original equation:** $$\frac{2}{3} x + 7 \frac{5}{6} = \frac{1}{2} x + 9$$ 3. **Step: Subtract $$\frac{1}{2} x$$ from both sides:** $$\frac{2}{3} x - \frac{1}{2} x + 7 \frac{5}{6} = 9$$ 4. **Combine like terms $$\frac{2}{3} x - \frac{1}{2} x$$:** Find common denominator 6: $$\frac{4}{6} x - \frac{3}{6} x = \frac{1}{6} x$$ So the equation becomes: $$\frac{1}{6} x + 7 \frac{5}{6} = 9$$ 5. **Subtract $$7 \frac{5}{6}$$ from both sides:** $$\frac{1}{6} x = 9 - 7 \frac{5}{6}$$ Convert mixed number to improper fraction: $$7 \frac{5}{6} = \frac{42}{6} + \frac{5}{6} = \frac{47}{6}$$ Convert 9 to fraction with denominator 6: $$9 = \frac{54}{6}$$ Subtract: $$\frac{54}{6} - \frac{47}{6} = \frac{7}{6}$$ So: $$\frac{1}{6} x = \frac{7}{6}$$ 6. **Multiply both sides by 6 to solve for $$x$$:** $$6 \times \frac{1}{6} x = 6 \times \frac{7}{6}$$ Use cancellation: $$\cancel{6} \times \frac{1}{\cancel{6}} x = \cancel{6} \times \frac{7}{\cancel{6}}$$ Simplifies to: $$x = 7$$ 7. **Check the student's errors:** - The student incorrectly wrote $$\frac{1}{6} x = 1 \frac{1}{6}$$ instead of $$\frac{1}{6} x = \frac{7}{6}$$. - The student then multiplied incorrectly and concluded $$x=1$$ instead of $$x=7$$. - The student also wrote $$x=9$$ at the end, which is incorrect. - The distributive property was not needed here, so no error there. - The main error is in arithmetic with fractions and combining like terms. **Final answer:** $$x=7$$