What is the value of $x$ in the equation $\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3 ?$
A. $-24$
B. $-6$
C. $-\frac{2}{3}$
D. 0

- First, distribute the constants on both sides of the equation to simplify it to \(\frac{1}{3} x + 8 = \frac{1}{6} x + 4\). - Next, combine the \(x\) terms by subtracting \(\frac{1}{6} x\) from both sides, resulting in \(\frac{1}{6} x + 8 = 4\). - Then, isolate the \(x\) term by subtracting \(8\) from both sides, which gives \(\frac{1}{6} x = -4\). - Finally, solve for \(x\) by multiplying both sides by \(6\), yielding \(x = \boxed{{-24}}\) . ### Explanation 1. Analyzing the Problem and Setting Up the Equation We are given a linear equation in one variable, \(x\): \[\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3.\] Our objective is to find the value of \(x\) that satisfies this equation. The solution plan involves simplifying both sides of the equation, combining like terms, isolating \(x\), and then solving for \(x\). 2. Distributing Constants and Simplifying Both Sides First, we distribute the constants into the parentheses on both sides of the equation to eliminate them. For the left side: \[\frac{2}{3}\left(\frac{1}{2} x+12\right) = \frac{2}{3} \times \frac{1}{2} x + \frac{2}{3} \times 12 = \frac{1}{3} x + 8.\] For the right side: \[\frac{1}{2}\left(\frac{1}{3} x+14\right)-3 = \frac{1}{2} \times \frac{1}{3} x + \frac{1}{2} \times 14 - 3 = \frac{1}{6} x + 7 - 3 = \frac{1}{6} x + 4.\] Now, the equation simplifies to: \[\frac{1}{3} x + 8 = \frac{1}{6} x + 4.\] 3. Isolating the Variable Term To isolate the terms containing \(x\) on one side, we subtract \(\frac{1}{6} x\) from both sides of the equation: \[\frac{1}{3} x - \frac{1}{6} x + 8 = 4.\] To combine the \(x\) terms, we find a common denominator, which is 6: \[\frac{2}{6} x - \frac{1}{6} x + 8 = 4.\] This simplifies to: \[\frac{1}{6} x + 8 = 4.\] 4. Solving for x Next, we isolate the constant terms on the other side by subtracting 8 from both sides of the equation: \[\frac{1}{6} x = 4 - 8.\] \[\frac{1}{6} x = -4.\] Finally, to solve for \(x\), we multiply both sides of the equation by 6: \[x = -4 \times 6.\] \[x = -24.\] Thus, the value of \(x\) that satisfies the given equation is \(-24\). 5. Conclusion The value of \(x\) that satisfies the equation \(\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3 \) is \(\boxed{{-24}}\) . ### Examples Solving linear equations is a fundamental skill used in many real-world scenarios. For instance, if you're managing a budget, you might use a linear equation to determine how many hours you need to work at a certain wage to reach a savings goal after accounting for expenses. Or, in physics, you might use a linear equation to calculate the time it takes for an object to travel a certain distance at a constant speed. These equations provide a structured way to find unknown values based on known relationships.