91Ó°ÊÓ

When working with fractions, it's important to have a common denominator in order to combine them. A common denominator is a shared multiple of the denominators of the fractions in question. In this exercise, we have the fractions \(\frac{u}{6}\) and \(\frac{u}{9}\). The smallest common multiple of 6 and 9 is 18. This means we need to rewrite the fractions so they have the same denominator (18). To do this, you find an equivalent fraction where the new denominator is the common denominator. For \(\frac{u}{6}\), multiply the numerator and the denominator by 3: \(\frac{3u}{18}\). For \(\frac{u}{9}\), multiply the numerator and the denominator by 2: \(\frac{2u}{18}\). Now both fractions share a common denominator, which allows us to combine or subtract them easily.

Solving linear equations is a fundamental concept in algebra. In this exercise, the linear equation is \(\frac{u}{18} = 4\). To solve for \u\, you want to isolate the variable on one side of the equation. You can achieve this by performing inverse operations. Here, \u\ is divided by 18. To isolate \u\, multiply both sides of the equation by 18: \(\frac{u}{18} \times 18 = 4 \times 18\). This simplifies to \u = 72\. Thus, \u\ is 72, indicating the solution to the equation. Consistently, these steps apply to various linear equations: isolate the variable by undoing the operations around it.

Subtracting fractions involves more than just subtracting their numerators and denominators directly. For the fractions \(\frac{3u}{18} - \frac{2u}{18}\), they already share a common denominator (18). This means we can subtract the numerators directly while keeping the same denominator. \(\frac{3u - 2u}{18}\) equals \(\frac{u}{18}\). Simplifying it further, we see that the subtraction does not change the denominator but simply reduces the fraction to a simpler form. Understanding how to handle the numerators while maintaining the common denominator is key to correctly subtracting fractions.