Solve the Linear Equation Involving Fractions: Finding X in 6x - 36/80 + 15x = -8x + 19/20

Let's solve for $x$ in the equation $6x - \frac{36}{80} + 15x = -8x + \frac{19}{20}$.

• Step 1: Simplify the equation by combining like terms on the left side. We have: $6x + 15x - \frac{36}{80} = -8x + \frac{19}{20}$ This becomes: $21x - \frac{36}{80} = -8x + \frac{19}{20}$
• Step 2: Simplify $\frac{36}{80}$ to its simplest form. The greatest common divisor of 36 and 80 is 4: $\frac{36}{80} = \frac{9}{20}$ Thus, the equation is now: $21x - \frac{9}{20} = -8x + \frac{19}{20}$
• Step 3: To eliminate fractions, multiply the entire equation by 20, the least common denominator, to clear the fractions: $20(21x) - 20 \left(\frac{9}{20}\right) = 20(-8x) + 20 \left(\frac{19}{20}\right)$ Simplifying gives: $420x - 9 = -160x + 19$
• Step 4: Combine like terms by adding $160x$ to both sides to collect $x$ terms on one side: $420x + 160x - 9 = 19$ This simplifies to: $580x - 9 = 19$
• Step 5: Isolate $x$ by adding 9 to both sides: $580x = 28$
• Step 6: Divide both sides by 580 to solve for $x$: $x = \frac{28}{580}$ Which simplifies to: $x = \frac{7}{145}$

Thus, the solution to $6x-\frac{36}{80}+15x=-8x+\frac{19}{20}$ is $x = \frac{7}{145}$.