Step 1

To find value of 'a' such that the system of equations 2x+3y=124x+ay=16 is inconsistent.

Using this value of 'a', to find value of 'b' such that the system of equations 2x+3y=124x+ay=b is dependent.

Step 2

Let the system of equations be

\(\displaystyle{a}_{{1}}{x}+{b}_{{1}}{y}+{c}_{{1}}={0}\)

\(\displaystyle{a}_{{2}}{x}+{b}_{{2}}{y}+{c}_{{2}}={0}\)

\(\displaystyle\frac{{a}_{{1}}}{{a}_{{2}}}=\frac{{b}_{{1}}}{{b}_{{2}}}\ne\frac{{c}_{{1}}}{{c}_{{2}}}\)

Thus, for given system of equations

\(\displaystyle\frac{{2}}{{4}}=\frac{{3}}{{a}}\ne-\frac{{12}}{{-{{16}}}}\)

\(\displaystyle\frac{{1}}{{2}}=\frac{{3}}{{a}}\)

a=6

Step 3

System of equations is dependent, if

\(\displaystyle\frac{{a}_{{1}}}{{a}_{{2}}}=\frac{{b}_{{1}}}{{b}_{{2}}}=\frac{{c}_{{1}}}{{c}_{{2}}}\)

Thus, for given system of equations

\(\displaystyle\frac{{2}}{{4}}=\frac{{3}}{{a}}=-\frac{{12}}{{-{{b}}}}\)

\(\displaystyle=\frac{{3}}{{6}}=\frac{{12}}{{b}}\)

\(\displaystyle\frac{{1}}{{2}}=\frac{{12}}{{b}}\)

b=24

To find value of 'a' such that the system of equations 2x+3y=124x+ay=16 is inconsistent.

Using this value of 'a', to find value of 'b' such that the system of equations 2x+3y=124x+ay=b is dependent.

Step 2

Let the system of equations be

\(\displaystyle{a}_{{1}}{x}+{b}_{{1}}{y}+{c}_{{1}}={0}\)

\(\displaystyle{a}_{{2}}{x}+{b}_{{2}}{y}+{c}_{{2}}={0}\)

\(\displaystyle\frac{{a}_{{1}}}{{a}_{{2}}}=\frac{{b}_{{1}}}{{b}_{{2}}}\ne\frac{{c}_{{1}}}{{c}_{{2}}}\)

Thus, for given system of equations

\(\displaystyle\frac{{2}}{{4}}=\frac{{3}}{{a}}\ne-\frac{{12}}{{-{{16}}}}\)

\(\displaystyle\frac{{1}}{{2}}=\frac{{3}}{{a}}\)

a=6

Step 3

System of equations is dependent, if

\(\displaystyle\frac{{a}_{{1}}}{{a}_{{2}}}=\frac{{b}_{{1}}}{{b}_{{2}}}=\frac{{c}_{{1}}}{{c}_{{2}}}\)

Thus, for given system of equations

\(\displaystyle\frac{{2}}{{4}}=\frac{{3}}{{a}}=-\frac{{12}}{{-{{b}}}}\)

\(\displaystyle=\frac{{3}}{{6}}=\frac{{12}}{{b}}\)

\(\displaystyle\frac{{1}}{{2}}=\frac{{12}}{{b}}\)

b=24