What is the solution set to the equation (2)/(7-m)=(4)/(m)-(5-m)/(7-m) A) 4,7 B) 4,5 C) 1,7 D) 4

To solve the equation $\frac{2}{7-m} = \frac{4}{m} - \frac{5-m}{7-m}$, we can start by finding a common denominator for the fractions on the right side of the equation. The common denominator is $7-m$. Multiplying both sides of the equation by $7-m$, we get:<br /><br />$2 = \frac{4(7-m)}{m} - (5-m)$<br /><br />Expanding and simplifying, we have:<br /><br />$2 = \frac{28-4m}{m} - 5 + m$<br /><br />Multiplying through by $m$ to clear the fraction, we get:<br /><br />$2m = 28 - 4m - 5m + m^2$<br /><br />Rearranging terms, we have:<br /><br />$m^2 - 9m + 28 = 0$<br /><br />This is a quadratic equation, which can be solved using the quadratic formula or by factoring. Factoring this equation, we get:<br /><br />$(m-4)(m-7) = 0$<br /><br />Setting each factor equal to zero, we have:<br /><br />$m-4 = 0$ or $m-7 = 0$<br /><br />Solving for $m$, we get:<br /><br />$m = 4$ or $m = 7$<br /><br />However, we must check these solutions in the original equation to ensure they are not extraneous. Plugging $m = 4$ into the original equation, we find that it is a valid solution. Plugging $m = 7$ into the original equation, we find that it is not a valid solution because it makes the denominator of the first fraction equal to zero.<br /><br />Therefore, the solution set to the equation is $\{4\}$, which corresponds to option D.