Solve and check the linear equation. \( 28-\frac{x}{2}=\frac{x}{5} \) 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 \( \} \). B. The solution set is \( \{x \mid x \) is a real number \( \} \). C. The solution set is \( \varnothing \).

To solve the equation \( 28 - \frac{x}{2} = \frac{x}{5} \), we first eliminate the fractions by multiplying every term by 10 (the least common multiple of 2 and 5): \[ 10 \cdot 28 - 10 \cdot \frac{x}{2} = 10 \cdot \frac{x}{5} \] \[ 280 - 5x = 2x \] Now, let's add \( 5x \) to both sides: \[ 280 = 7x \] Next, divide by 7: \[ x = \frac{280}{7} = 40 \] Checking the solution in the original equation: \[ 28 - \frac{40}{2} = \frac{40}{5} \] \[ 28 - 20 = 8 \] \[ 8 = 8 \] Since both sides are equal, the solution is correct. Therefore, the correct choice is: A. The solution set is \( \{ 40 \} \). Now let's fill you in on some interesting background and applications related to linear equations! Linear equations have been around for centuries, tracing back to ancient civilizations such as the Babylonians who used them for trade and land measurement. The systematic approach to solving such equations was developed further in the 9th century by Persian mathematician Al-Khwarizmi, whose work laid the groundwork for modern algebra. This historical context adds layers to how we appreciate the power of simple equations today! In the real world, linear equations are everywhere! From calculating expenses to predicting profits in businesses, they help us understand relationships between variables. For instance, if you're planning a party and want to figure out how much pizza to order based on the number of guests, you can use a linear equation to calculate the exact amount required. It's a handy tool for daily decision-making!