To solve the equation [tex]\( 2 = \frac{2}{3} - \frac{5}{y} \)[/tex] for [tex]\( y \)[/tex], we will follow these steps:
1. Isolate the fraction involving [tex]\( y \)[/tex]:
[tex]\[ 2 = \frac{2}{3} - \frac{5}{y} \][/tex]
To isolate the term involving [tex]\( y \)[/tex], we can start by isolating [tex]\(\frac{5}{y}\)[/tex]. Subtract [tex]\(\frac{2}{3}\)[/tex] from both sides of the equation:
[tex]\[ 2 - \frac{2}{3} = -\frac{5}{y} \][/tex]
2. Simplify the left side of the equation:
[tex]\[ 2 - \frac{2}{3} \][/tex]
To perform this subtraction, we need a common denominator. The common denominator for 2 and [tex]\(\frac{2}{3}\)[/tex] is 3. Rewrite 2 as [tex]\(\frac{6}{3}\)[/tex]:
[tex]\[ \frac{6}{3} - \frac{2}{3} = \frac{4}{3} \][/tex]
So the equation now becomes:
[tex]\[ \frac{4}{3} = -\frac{5}{y} \][/tex]
3. Solve for [tex]\( y \)[/tex]:
We have [tex]\(\frac{4}{3} = -\frac{5}{y} \)[/tex]. To eliminate the fraction, we can cross-multiply:
[tex]\[ 4y = -5 \cdot 3 \][/tex]
Simplify the right side:
[tex]\[ 4y = -15 \][/tex]
4. Isolate [tex]\( y \)[/tex]:
Divide both sides of the equation by 4:
[tex]\[ y = \frac{-15}{4} \][/tex]
So, the solution for [tex]\( y \)[/tex] is:
[tex]\[ y = -\frac{15}{4} \][/tex]
In conclusion, the value of [tex]\( y \)[/tex] that satisfies the equation [tex]\( 2 = \frac{2}{3} - \frac{5}{y} \)[/tex] is [tex]\( y = -\frac{15}{4} \)[/tex].
