Question

1. $\frac{4}{5}+\frac{5}{y - 3}=\frac{5}{5y-15}$

Explanation:

Step1: Find a common denominator

The common - denominator of 5, \(y - 3\) and \(5y-15\) is \(5(y - 3)\). Rewrite the equation \(\frac{4}{5}+\frac{5}{y - 3}=\frac{5}{5y-15}\) as \(\frac{4(y - 3)}{5(y - 3)}+\frac{5\times5}{5(y - 3)}=\frac{5}{5(y - 3)}\).
Since \(5y-15 = 5(y - 3)\).

Step2: Simplify the left - hand side

\(\frac{4(y - 3)+25}{5(y - 3)}=\frac{5}{5(y - 3)}\). Expand \(4(y - 3)\) to get \(4y-12\). Then the left - hand side is \(\frac{4y-12 + 25}{5(y - 3)}=\frac{4y + 13}{5(y - 3)}\).
So, \(\frac{4y+13}{5(y - 3)}=\frac{5}{5(y - 3)}\).

Step3: Cross - multiply (since denominators are non - zero, \(y

eq3\))
Since the denominators are the same and non - zero, we can set the numerators equal: \(4y+13 = 5\).

Step4: Solve for \(y\)

Subtract 13 from both sides: \(4y=5 - 13=-8\). Then divide both sides by 4: \(y=\frac{-8}{4}=-2\).

Answer:

\(y=-2\)