Question - Solving a Linear Equation with Fractions
Solution:
The equation in the image is:$$ \frac{1}{2}(3x + 6) - \frac{1}{3}(2x - 4) = 20 $$To solve this equation for x, follow these steps:1. Distribute the fractions to the terms inside the parentheses:$$ \frac{1}{2} \cdot 3x + \frac{1}{2} \cdot 6 - \frac{1}{3} \cdot 2x + \frac{1}{3} \cdot 4 = 20 $$2. Simplify the distributed terms:$$ \frac{3}{2}x + 3 - \frac{2}{3}x + \frac{4}{3} = 20 $$3. Combine like terms:$$ \frac{3}{2}x - \frac{2}{3}x = 20 - 3 - \frac{4}{3} $$4. To combine the x terms, find a common denominator, which in this case is 6:$$ \frac{9}{6}x - \frac{4}{6}x = 17 - \frac{4}{3} $$5. Simplify the x terms and convert 17 into a fraction with the same denominator as 4/3 to continue simplifying:$$ \frac{5}{6}x = \frac{51}{3} - \frac{4}{3} $$6. Simplify the right side of the equation:$$ \frac{5}{6}x = \frac{47}{3} $$7. To solve for x, multiply by the reciprocal of 5/6, which is 6/5:$$ x = \frac{47}{3} \cdot \frac{6}{5} $$8. Multiply the two fractions:$$ x = \frac{47 \cdot 6}{3 \cdot 5} $$9. Simplify the multiplication:$$ x = \frac{282}{15} $$10. Finally, simplify the fraction if possible:$$ x = \frac{47 \cdot 2}{3 \cdot 5} $$$$ x = \frac{94}{15} $$$$ x = 6 \frac{4}{15} $$So, the solution to the equation is $$ x = 6 \frac{4}{15} $$.
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