5) \( \frac{1}{5} x+\frac{13}{7}\left(x+\frac{7}{6}\right)=\frac{11}{6} x+\frac{821}{350} \)

To solve the equation \( \frac{1}{5} x+\frac{13}{7}\left(x+\frac{7}{6}\right)=\frac{11}{6} x+\frac{821}{350} \), we start by simplifying the left-hand side. First, distribute \( \frac{13}{7} \) in the term \( \frac{13}{7}\left(x+\frac{7}{6}\right) \): \[ \frac{13}{7} x + \frac{13}{7} \cdot \frac{7}{6} = \frac{13}{7} x + \frac{13}{6} \] Now, rewrite the equation: \[ \frac{1}{5} x + \frac{13}{7} x + \frac{13}{6} = \frac{11}{6} x + \frac{821}{350} \] Next, we need a common denominator to combine the x-terms on the left side. The common denominator for \( 5 \) and \( 7 \) is \( 35 \): \[ \frac{1}{5} x = \frac{7}{35} x, \quad \frac{13}{7} x = \frac{65}{35} x \] Combining these gives: \[ \frac{1}{5} x + \frac{13}{7} x = \left( \frac{7}{35} + \frac{65}{35} \right)x = \frac{72}{35} x \] Now, rewrite the left side with this combination: \[ \frac{72}{35} x + \frac{13}{6} = \frac{11}{6} x + \frac{821}{350} \] Next, let's isolate the x-terms. Subtract \( \frac{11}{6} x \) from both sides: \[ \frac{72}{35} x - \frac{11}{6} x + \frac{13}{6} = \frac{821}{350} \] Now find a common denominator for \( \frac{72}{35} \) and \( \frac{11}{6} \). The least common multiple of \( 35 \) and \( 6 \) is \( 210 \): \[ \frac{72}{35} = \frac{72 \times 6}{35 \times 6} = \frac{432}{210}, \quad \frac{11}{6} = \frac{11 \times 35}{6 \times 35} = \frac{385}{210} \] Now, substituting those values back into the equation gives: \[ \frac{432}{210} x - \frac{385}{210} x + \frac{13}{6} = \frac{821}{350} \] Combine the x-terms: \[ \left( \frac{432 - 385}{210} \right)x + \frac{13}{6} = \frac{821}{350} \] This simplifies to: \[ \frac{47}{210} x + \frac{13}{6} = \frac{821}{350} \] Next, to isolate \( x \), we’ll subtract \( \frac{13}{6} \) from both sides. Convert \( \frac{13}{6} \) to have a denominator of \( 210 \): \[ \frac{13}{6} = \frac{13 \times 35}{6 \times 35} = \frac{455}{210} \] Now, substituting this into the equation gives: \[ \frac{47}{210} x = \frac{821}{350} - \frac{455}{210} \] Convert \( \frac{821}{350} \) into a fraction with a denominator of \( 210 \): \[ \frac{821}{350} = \frac{821 \times 6}{350 \times 6} = \frac{4926}{2100} \quad (where \, 2100 \, is \, lcm(210,350)) \] Finding a common denominator for \( 450 \) and \( 2100 \) results in: \[ \frac{455}{210} = \frac{455 \times 10}{210 \times 10} = \frac{4550}{2100} \] So we now find: \[ \frac{821}{350} = \frac{4926}{2100} - \frac{4550}{2100} = \frac{4926 - 4550}{2100} = \frac{376}{2100} \] Now multiplying