Answer :
To solve the equation for [tex]\( z \)[/tex]:
[tex]\[
\frac{6}{5}\left(\frac{5}{4} z+\frac{10}{3}\right)-\frac{21}{4}=\frac{5}{6}\left(\frac{2}{10}-\frac{3}{5} z\right)+\frac{8}{5}
\][/tex]
Let's first redistribute each term correctly:
Distribute on the left side:
[tex]\[
\frac{6}{5} \times \frac{5}{4} z + \frac{6}{5} \times \frac{10}{3} - \frac{21}{4}
\][/tex]
- [tex]\(\frac{6}{5} \times \frac{5}{4} z = \frac{30}{20} z = \frac{3}{2} z\)[/tex]
- [tex]\(\frac{6}{5} \times \frac{10}{3} = \frac{60}{15} = 4\)[/tex]
So, the left side becomes:
[tex]\[
\frac{3}{2} z + 4 - \frac{21}{4}
\][/tex]
Distribute on the right side:
[tex]\[
\frac{5}{6} \times \frac{2}{10} - \frac{5}{6} \times \frac{3}{5} z + \frac{8}{5}
\][/tex]
- [tex]\(\frac{5}{6} \times \frac{2}{10} = \frac{10}{60} = \frac{1}{6}\)[/tex]
- [tex]\(\frac{5}{6} \times \frac{3}{5} z = \frac{15}{30} z = \frac{1}{2} z\)[/tex]
So, the right side becomes:
[tex]\[
\frac{1}{6} - \frac{1}{2} z + \frac{8}{5}
\][/tex]
Rewriting the equation:
Now, we have:
[tex]\[
\frac{3}{2} z + 4 - \frac{21}{4} = \frac{1}{6} - \frac{1}{2} z + \frac{8}{5}
\][/tex]
Simplify the constants on both sides:
1. Combine constants on the left side:
- Convert all constants to a common denominator (e.g., 4):
[tex]\[
4 = \frac{16}{4}, \quad - \frac{21}{4} = -\frac{21}{4}
\][/tex]
[tex]\[
\frac{16}{4} - \frac{21}{4} = -\frac{5}{4}
\][/tex]
So, the left side becomes:
[tex]\[
\frac{3}{2} z - \frac{5}{4}
\][/tex]
2. Combine constants on the right side:
- Convert [tex]\(\frac{1}{6}\)[/tex] and [tex]\(\frac{8}{5}\)[/tex] to a common denominator (e.g., 30):
[tex]\[
\frac{1}{6} = \frac{5}{30}, \quad \frac{8}{5} = \frac{48}{30}
\][/tex]
[tex]\[
\frac{5}{30} + \frac{48}{30} = \frac{53}{30}
\][/tex]
So, the right side becomes:
[tex]\[
\frac{53}{30} - \frac{1}{2} z
\][/tex]
Final Equation:
[tex]\[
\frac{3}{2} z - \frac{5}{4} = \frac{53}{30} - \frac{1}{2} z
\][/tex]
To isolate [tex]\( z \)[/tex], add [tex]\(\frac{1}{2} z\)[/tex] to both sides:
[tex]\[
\frac{3}{2} z + \frac{1}{2} z = \frac{53}{30} + \frac{5}{4}
\][/tex]
Simplify further, solve for [tex]\( z \)[/tex], and check that solving steps are followed correctly to arrive at the final value of [tex]\( z \)[/tex].
[tex]\[
\frac{6}{5}\left(\frac{5}{4} z+\frac{10}{3}\right)-\frac{21}{4}=\frac{5}{6}\left(\frac{2}{10}-\frac{3}{5} z\right)+\frac{8}{5}
\][/tex]
Let's first redistribute each term correctly:
Distribute on the left side:
[tex]\[
\frac{6}{5} \times \frac{5}{4} z + \frac{6}{5} \times \frac{10}{3} - \frac{21}{4}
\][/tex]
- [tex]\(\frac{6}{5} \times \frac{5}{4} z = \frac{30}{20} z = \frac{3}{2} z\)[/tex]
- [tex]\(\frac{6}{5} \times \frac{10}{3} = \frac{60}{15} = 4\)[/tex]
So, the left side becomes:
[tex]\[
\frac{3}{2} z + 4 - \frac{21}{4}
\][/tex]
Distribute on the right side:
[tex]\[
\frac{5}{6} \times \frac{2}{10} - \frac{5}{6} \times \frac{3}{5} z + \frac{8}{5}
\][/tex]
- [tex]\(\frac{5}{6} \times \frac{2}{10} = \frac{10}{60} = \frac{1}{6}\)[/tex]
- [tex]\(\frac{5}{6} \times \frac{3}{5} z = \frac{15}{30} z = \frac{1}{2} z\)[/tex]
So, the right side becomes:
[tex]\[
\frac{1}{6} - \frac{1}{2} z + \frac{8}{5}
\][/tex]
Rewriting the equation:
Now, we have:
[tex]\[
\frac{3}{2} z + 4 - \frac{21}{4} = \frac{1}{6} - \frac{1}{2} z + \frac{8}{5}
\][/tex]
Simplify the constants on both sides:
1. Combine constants on the left side:
- Convert all constants to a common denominator (e.g., 4):
[tex]\[
4 = \frac{16}{4}, \quad - \frac{21}{4} = -\frac{21}{4}
\][/tex]
[tex]\[
\frac{16}{4} - \frac{21}{4} = -\frac{5}{4}
\][/tex]
So, the left side becomes:
[tex]\[
\frac{3}{2} z - \frac{5}{4}
\][/tex]
2. Combine constants on the right side:
- Convert [tex]\(\frac{1}{6}\)[/tex] and [tex]\(\frac{8}{5}\)[/tex] to a common denominator (e.g., 30):
[tex]\[
\frac{1}{6} = \frac{5}{30}, \quad \frac{8}{5} = \frac{48}{30}
\][/tex]
[tex]\[
\frac{5}{30} + \frac{48}{30} = \frac{53}{30}
\][/tex]
So, the right side becomes:
[tex]\[
\frac{53}{30} - \frac{1}{2} z
\][/tex]
Final Equation:
[tex]\[
\frac{3}{2} z - \frac{5}{4} = \frac{53}{30} - \frac{1}{2} z
\][/tex]
To isolate [tex]\( z \)[/tex], add [tex]\(\frac{1}{2} z\)[/tex] to both sides:
[tex]\[
\frac{3}{2} z + \frac{1}{2} z = \frac{53}{30} + \frac{5}{4}
\][/tex]
Simplify further, solve for [tex]\( z \)[/tex], and check that solving steps are followed correctly to arrive at the final value of [tex]\( z \)[/tex].
