To determine the value of [tex]\( y \)[/tex] for which the expression [tex]\(\frac{5}{\frac{3}{4} y - \frac{5}{8}}\)[/tex] has no solutions, we need to find when the denominator is equal to zero. If the denominator is zero, the expression becomes undefined because division by zero is not possible.
1. Start with the denominator of the expression:
[tex]\[
\frac{3}{4} y - \frac{5}{8}
\][/tex]
2. Set the denominator equal to zero to find the critical value of [tex]\( y \)[/tex]:
[tex]\[
\frac{3}{4} y - \frac{5}{8} = 0
\][/tex]
3. Solve for [tex]\( y \)[/tex].
Step 1: Move [tex]\(\frac{5}{8}\)[/tex] to the other side of the equation:
[tex]\[
\frac{3}{4} y = \frac{5}{8}
\][/tex]
Step 2: To isolate [tex]\( y \)[/tex], we need to divide both sides by [tex]\(\frac{3}{4}\)[/tex]. This is equivalent to multiplying both sides by the reciprocal of [tex]\(\frac{3}{4}\)[/tex], which is [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[
y = \frac{5}{8} \times \frac{4}{3}
\][/tex]
Step 3: Perform the multiplication on the right side:
[tex]\[
y = \frac{5 \cdot 4}{8 \cdot 3} = \frac{20}{24}
\][/tex]
Step 4: Simplify the fraction [tex]\(\frac{20}{24}\)[/tex]:
[tex]\[
y = \frac{5}{6}
\][/tex]
So, the value of [tex]\( y \)[/tex] that makes the denominator zero and thus makes the expression [tex]\(\frac{5}{\frac{3}{4} y - \frac{5}{8}}\)[/tex] have no solutions is:
[tex]\[
\boxed{\frac{5}{6}}
\][/tex]
