Question

solve for u.
\\(\frac{2}{3}u = \frac{1}{2} + \frac{5}{3}u + \frac{5}{2}u - 2\\)
\\(u = \\)
submit

Explanation:

Step1: Simplify the right side (combine like terms)

First, combine the \(u\)-terms on the right: \(\frac{5}{3}u+\frac{5}{2}u\). Find a common denominator, which is 6. So \(\frac{5}{3}u=\frac{10}{6}u\) and \(\frac{5}{2}u = \frac{15}{6}u\). Then \(\frac{10}{6}u+\frac{15}{6}u=\frac{25}{6}u\). Next, combine the constant terms: \(\frac{1}{2}-2\). \(-2=-\frac{4}{2}\), so \(\frac{1}{2}-\frac{4}{2}=-\frac{3}{2}\). Now the equation is \(\frac{2}{3}u=-\frac{3}{2}+\frac{25}{6}u\).

Step2: Get all \(u\)-terms on one side

Subtract \(\frac{25}{6}u\) from both sides. First, rewrite \(\frac{2}{3}u\) with denominator 6: \(\frac{2}{3}u=\frac{4}{6}u\). So \(\frac{4}{6}u-\frac{25}{6}u=-\frac{3}{2}\). Combining the \(u\)-terms: \(\frac{4 - 25}{6}u=-\frac{3}{2}\), which is \(-\frac{21}{6}u=-\frac{3}{2}\). Simplify \(-\frac{21}{6}u\) to \(-\frac{7}{2}u\).

Step3: Solve for \(u\)

Now we have \(-\frac{7}{2}u=-\frac{3}{2}\). Multiply both sides by \(-\frac{2}{7}\) (the reciprocal of \(-\frac{7}{2}\)) to isolate \(u\). So \(u = (-\frac{3}{2})\times(-\frac{2}{7})\). The \(-2\) in the numerator and denominator cancels, leaving \(u=\frac{3}{7}\).

Answer:

\(\frac{3}{7}\)