Solve and check the linear equation. \( \frac{x+6}{14}=\frac{4}{7}+\frac{x-5}{6} \) What is the solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \} \). (Simplify your answer. Type an integer or a fraction.) B. The solution set is \( \{x \mid x \) is a real number \( \} \). C. The solution set is \( \varnothing \).

To solve the equation \( \frac{x+6}{14} = \frac{4}{7} + \frac{x-5}{6} \), we start by simplifying both sides. First, we convert all the fractions to have a common denominator. The common denominator for 14, 7, and 6 is 42. Rewriting the equation: \[ \frac{x+6}{14} = \frac{3(4)}{42} + \frac{7(x-5)}{42} \] This becomes: \[ \frac{x+6}{14} = \frac{12 + 7x - 35}{42} \] Next, multiplying both sides by 42 to eliminate the denominators gives: \[ 3(x + 6) = 12 + 7x - 35 \] Distributing the left side: \[ 3x + 18 = 7x - 23 \] Next, we rearrange the equation to isolate \( x \): \[ 18 + 23 = 7x - 3x \] \[ 41 = 4x \] Then, divide both sides by 4: \[ x = \frac{41}{4} \] Now we have our solution: \( x = \frac{41}{4} \). To check our solution, we substitute \( \frac{41}{4} \) back into the original equation: Left side: \[ \frac{\frac{41}{4} + 6}{14} = \frac{\frac{41}{4} + \frac{24}{4}}{14} = \frac{\frac{65}{4}}{14} = \frac{65}{56} \] Right side: Calculate \( \frac{4}{7} + \frac{x-5}{6} \) at \( x = \frac{41}{4} \): \[ x - 5 = \frac{41}{4} - 5 = \frac{41}{4} - \frac{20}{4} = \frac{21}{4} \] Now substitute: \[ \frac{4}{7} + \frac{21/4}{6} = \frac{4}{7} + \frac{21}{24} = \frac{96}{168} + \frac{49}{168} = \frac{145}{168} \] We see that both sides are equal; thus, \( x = \frac{41}{4} \) is indeed the solution. The correct choice is: A. The solution set is \( \{ \frac{41}{4} \} \). --- A little fun fact: The world of linear equations goes back thousands of years! Ancient Babylonians used linear equations to solve problems related to trade and taxation long before the modern notation we use today was developed. It’s amazing how these concepts have withstood the test of time! Feeling stuck? Here’s a quick tip: Always combine like terms and simplify as much as possible during each step! A common mistake is losing track of negative signs or forgetting to distribute! Keeping your work neat can save you from unnecessary errors.