Calculation:

Consider the given equation, \(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}-{2}\right]}={5}{y}+{6}{\left({7}+{y}\right)}-{3}\)

Now, apply the distributive property:

\(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}-{2}\right]}={5}{y}+{6}{\left({7}+{9}\right)}-{3}\)

\(\displaystyle{3}{y}+{2}{\left[{5}{y}—{20}-{2}\right]}={5}{y}+{42}+{6}{y}—{3}\)

\(\displaystyle{3}{y}+{10}{y}—{44}={5}{y}+{42}+{6}{y}—{3}\)

Combine the like terms:

\(\displaystyle{3}{y}+{10}{y}—{44}={5}{y}+{42}+{6}{y}—{3}\)

\(\displaystyle{13}{y}-{44}={11}{y}+{39}\)

Now, subtract I ly from both sides:

\(\displaystyle{13}{y}-{44}={11}{y}+{39}\)

\(\displaystyle{13}{y}—{44}—{11}{y}={11}{y}+{39}+{11}{y}\)

Combine the like terms:

\(\displaystyle{13}{y}—{44}—{11}{y}={11}{y}+{39}+{11}{y}\)

\(\displaystyle{2}{y}-{44}={39}\)

Add 44 to both sides:

\(\displaystyle{2}{y}-{44}+{44}={39}+{44}\)

Combine the like terms:

\(\displaystyle{2}{y}={83}\)

Divide both sides by 2:

\(\displaystyle{y}={\frac{{{83}}}{{{2}}}}\)

Now, check the solution by put the value of y = © in the equation

\(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}—{2}\right]}={5}{y}+{6}{\left({7}+{9}\right)}-{3}\).

\(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}-{2}\right]}={5}{y}+{6}{\left({7}+{y}\right)}-{3}\)

\(\displaystyle{3}{\left({\frac{{{83}}}{{{2}}}}\right)}+{2}{\left[{5}{\left({\frac{{{83}}}{{{2}}}}-{4}\right)}-{2}\right]}{\overset{{?}}{{=}}}{5}{\left({\frac{{{83}}}{{{2}}}}\right)}+{6}{\left({7}+{\frac{{{83}}}{{{2}}}}\right)}-{3}\)

\(\displaystyle{\left({\frac{{{249}}}{{{2}}}}\right)}+{2}{\left[{\left({\frac{{{415}}}{{{2}}}}-{20}\right)}-{2}\right]}{\overset{{?}}{{=}}}{\left({\frac{{{415}}}{{{2}}}}\right)}+{\left({42}+{\frac{{{498}}}{{{2}}}}\right)}-{3}\)

\(\displaystyle{\left({\frac{{{249}}}{{{2}}}}\right)}+{\left[{\frac{{{830}}}{{{2}}}}-{40}-{4}\right]}{\overset{{?}}{{=}}}{\left({\frac{{{415}}}{{{2}}}}\right)}+{42}+{\frac{{{498}}}{{{2}}}}-{3}\)

On further solution:

\(\displaystyle{\frac{{{249}}}{{{2}}}}+{\frac{{{830}}}{{{2}}}}-{44}{\overset{{?}}{{=}}}{\frac{{{415}}}{{{2}}}}+{\frac{{{498}}}{{{2}}}}+{39}\)

\(\displaystyle{\frac{{{249}+{830}-{88}}}{{{2}}}}{\overset{{?}}{{=}}}{\frac{{{415}+{498}+{78}}}{{{2}}}}\)

\(\displaystyle{\frac{{{991}}}{{{2}}}}{\overset{{?}}{{=}}}{\frac{{{991}}}{{{2}}}}\)

Hence, the solution set is \(\displaystyle{y}={\frac{{{83}}}{{{2}}}}\).

Consider the given equation, \(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}-{2}\right]}={5}{y}+{6}{\left({7}+{y}\right)}-{3}\)

Now, apply the distributive property:

\(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}-{2}\right]}={5}{y}+{6}{\left({7}+{9}\right)}-{3}\)

\(\displaystyle{3}{y}+{2}{\left[{5}{y}—{20}-{2}\right]}={5}{y}+{42}+{6}{y}—{3}\)

\(\displaystyle{3}{y}+{10}{y}—{44}={5}{y}+{42}+{6}{y}—{3}\)

Combine the like terms:

\(\displaystyle{3}{y}+{10}{y}—{44}={5}{y}+{42}+{6}{y}—{3}\)

\(\displaystyle{13}{y}-{44}={11}{y}+{39}\)

Now, subtract I ly from both sides:

\(\displaystyle{13}{y}-{44}={11}{y}+{39}\)

\(\displaystyle{13}{y}—{44}—{11}{y}={11}{y}+{39}+{11}{y}\)

Combine the like terms:

\(\displaystyle{13}{y}—{44}—{11}{y}={11}{y}+{39}+{11}{y}\)

\(\displaystyle{2}{y}-{44}={39}\)

Add 44 to both sides:

\(\displaystyle{2}{y}-{44}+{44}={39}+{44}\)

Combine the like terms:

\(\displaystyle{2}{y}={83}\)

Divide both sides by 2:

\(\displaystyle{y}={\frac{{{83}}}{{{2}}}}\)

Now, check the solution by put the value of y = © in the equation

\(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}—{2}\right]}={5}{y}+{6}{\left({7}+{9}\right)}-{3}\).

\(\displaystyle{3}{y}+{2}{\left[{5}{\left({y}-{4}\right)}-{2}\right]}={5}{y}+{6}{\left({7}+{y}\right)}-{3}\)

\(\displaystyle{3}{\left({\frac{{{83}}}{{{2}}}}\right)}+{2}{\left[{5}{\left({\frac{{{83}}}{{{2}}}}-{4}\right)}-{2}\right]}{\overset{{?}}{{=}}}{5}{\left({\frac{{{83}}}{{{2}}}}\right)}+{6}{\left({7}+{\frac{{{83}}}{{{2}}}}\right)}-{3}\)

\(\displaystyle{\left({\frac{{{249}}}{{{2}}}}\right)}+{2}{\left[{\left({\frac{{{415}}}{{{2}}}}-{20}\right)}-{2}\right]}{\overset{{?}}{{=}}}{\left({\frac{{{415}}}{{{2}}}}\right)}+{\left({42}+{\frac{{{498}}}{{{2}}}}\right)}-{3}\)

\(\displaystyle{\left({\frac{{{249}}}{{{2}}}}\right)}+{\left[{\frac{{{830}}}{{{2}}}}-{40}-{4}\right]}{\overset{{?}}{{=}}}{\left({\frac{{{415}}}{{{2}}}}\right)}+{42}+{\frac{{{498}}}{{{2}}}}-{3}\)

On further solution:

\(\displaystyle{\frac{{{249}}}{{{2}}}}+{\frac{{{830}}}{{{2}}}}-{44}{\overset{{?}}{{=}}}{\frac{{{415}}}{{{2}}}}+{\frac{{{498}}}{{{2}}}}+{39}\)

\(\displaystyle{\frac{{{249}+{830}-{88}}}{{{2}}}}{\overset{{?}}{{=}}}{\frac{{{415}+{498}+{78}}}{{{2}}}}\)

\(\displaystyle{\frac{{{991}}}{{{2}}}}{\overset{{?}}{{=}}}{\frac{{{991}}}{{{2}}}}\)

Hence, the solution set is \(\displaystyle{y}={\frac{{{83}}}{{{2}}}}\).