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LinearEquationWithAllFractions

Solution to the Equation with Distributive Property

Consider the equation: \( \frac{2}{3}(4x – \frac{3}{5}) + \frac{5}{2}(2x + \frac{1}{4}) = \frac{1}{6}x + \frac{1}{2} \)

To solve this equation, we will apply the distributive property to simplify the expressions. Then, we’ll combine like terms and isolate \( x \) to find the solution.

Step-by-Step Solution:

  1. Apply the distributive property to both terms on the left-hand side of the equation: \( \frac{2}{3} \cdot 4x – \frac{2}{3} \cdot \frac{3}{5} + \frac{5}{2} \cdot 2x + \frac{5}{2} \cdot \frac{1}{4} = \frac{1}{6}x + \frac{1}{2} \)
  2. Simplify the expressions on both sides of the equation: \( \frac{8}{3}x – \frac{2}{5} + 5x + \frac{5}{8} = \frac{1}{6}x + \frac{1}{2} \)
  3. Combine like terms on both sides of the equation: \( \frac{23}{3}x – \frac{2}{5} + \frac{5}{8} = \frac{1}{6}x + \frac{1}{2} \)
  4. Bring \( x \) terms to one side by subtracting \( \frac{1}{6}x \) from both sides: \( \frac{23}{3}x – \frac{1}{6}x = \frac{1}{2} – \frac{5}{8} + \frac{2}{5} \)
  5. To add the fractions on the right-hand side, we find the common denominator, which is \( 120 \): \( \frac{110}{120}x = \frac{60 – 75 + 48}{120} \)
  6. Combine fractions on both sides of the equation: \( \frac{110}{120}x = \frac{33}{120} \)
  7. Solve for \( x \) by dividing both sides by \( \frac{110}{120} \): \( x = \frac{33}{120} \div \frac{110}{120} \)
  8. Calculate the result: \( x = \frac{33}{110} \)

Therefore, the solution to the equation is \( x = \frac{11}{300} \).

Solution to the Equation \( \frac{2}{3}x + \frac{1}{4} = \frac{5}{6} \)

To solve this equation of the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are all fractions, follow the steps below:

    1. Bring the constant term (\( \frac{1}{4} \)) to the other side by subtracting it from both sides:

\( \frac{2}{3}x = \frac{5}{6} – \frac{1}{4} \)

    1. Find the common denominator for the fractions on the right-hand side, which is 12:

\( \frac{2}{3}x = \frac{10}{12} – \frac{3}{12} \)

    1. Combine the fractions on the right-hand side:

\( \frac{2}{3}x = \frac{7}{12} \)

    1. Isolate \( x \) by dividing both sides by \( \frac{2}{3} \):

\( x = \frac{7}{12} \div \frac{2}{3} \)

    1. Invert the divisor and multiply:

\( x = \frac{7}{12} \times \frac{3}{2} \)

    1. Calculate the result:

\( x = \frac{21}{24} \)

    1. Simplify the fraction, if possible:

\( x = \frac{7}{8} \)

Therefore, the solution to the equation is \( x = \frac{7}{8} \).