College

Jared needs to solve for [tex]z[/tex] in the equation:

[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].

He distributed on both sides of the equation as shown:

Original Equation:
[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}{3}[/tex]

Distribution:
[tex]\frac{30}{20} z+\frac{e 0}{15}-\frac{31}{4}=\frac{45}{60}-\frac{15}{50} z+\frac{8}{3}[/tex]

What should Jared do next?

A. Add [tex]\frac{30}{20}[/tex] to both sides of the equation.
B. Subtract [tex]\frac{\frac{45}{6}}{6}-\frac{18}{80} z[/tex].
C. Add [tex]\frac{12}{4}[/tex] to both sides of the equation.
D. Subtract [tex]\frac{6}{15}-\frac{21}{4}[/tex].

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].