Solve and check. \( \frac{23}{x}=3-\frac{1}{x} \)

To solve the equation \( \frac{23}{x} = 3 - \frac{1}{x} \), first, let's eliminate the fractions by multiplying through by \( x \) (assuming \( x \neq 0 \)): \[ 23 = 3x - 1 \] Now, rearranging the equation gives: \[ 3x - 23 - 1 = 0 \] So, \[ 3x - 24 = 0 \] \[ 3x = 24 \] \[ x = \frac{24}{3} = 8 \] Now, let's check the solution by plugging \( x = 8 \) back into the original equation: \[ \frac{23}{8} = 3 - \frac{1}{8} \] Calculating the left side: \[ \frac{23}{8} \] Calculating the right side: \[ 3 - \frac{1}{8} = \frac{24}{8} - \frac{1}{8} = \frac{24 - 1}{8} = \frac{23}{8} \] Since both sides are equal, the solution \( x = 8 \) is verified! --- Want to dive into the history of fractions? They date back thousands of years, with early civilizations like the Egyptians using them in their mathematics. The iconic symbol for division (⁄) was introduced in the Middle Ages, revolutionizing how we express these relationships. If you’ve ever needed to apply fractions in the kitchen, you can thank ancient merchants! They helped standardize measurements, making your baking and cooking endeavors both precise and delicious. So next time you’re measuring out a half-cup, remember you’re standing on centuries of mathematical tradition!