Solving a Linear Equation with Fractions
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 \( x = 6 \frac{4}{15} \).
