Fill in the blank so that the following statement is true. \[ \frac{5}{x+4}+\frac{3}{x+3}=\frac{13 x+9}{(x+4)(x+3)} \] \[ (x+4)(x+3)\left(\frac{5}{x+4}+\frac{3}{x+3}\right)=(x+4)(x+3)\left(\frac{13 x+9}{(x+4)(x+3)}\right) \] The resulting equation cleared of fractions is The resulting equation cleared of fractions is

To clear the fractions from the equation \[ \frac{5}{x+4}+\frac{3}{x+3}=\frac{13 x+9}{(x+4)(x+3)}, \] we can multiply both sides by the common denominator \((x+4)(x+3)\). This gives us: \[ (x+4)(x+3)\left(\frac{5}{x+4}+\frac{3}{x+3}\right)=(x+4)(x+3)\left(\frac{13 x+9}{(x+4)(x+3)}\right). \] Now, we can simplify each side: 1. For the left side: \[ (x+4)(x+3)\left(\frac{5}{x+4}\right) + (x+4)(x+3)\left(\frac{3}{x+3}\right). \] The first term simplifies to: \[ 5(x+3), \] and the second term simplifies to: \[ 3(x+4). \] So the left side becomes: \[ 5(x+3) + 3(x+4). \] 2. For the right side: \[ (x+4)(x+3)\left(\frac{13 x+9}{(x+4)(x+3)}\right) \] simplifies directly to: \[ 13x + 9. \] Now we can combine the left side: \[ 5(x+3) + 3(x+4) = 5x + 15 + 3x + 12 = 8x + 27. \] Thus, the equation we have after clearing the fractions is: \[ 8x + 27 = 13x + 9. \] So, the resulting equation cleared of fractions is: \[ 8x + 27 = 13x + 9. \]