Step 1

we have the given two equations

\(\displaystyle{x}^{{2}}+{y}^{{2}}−{19}{x}+{y}+{66}={0}\)

x−y−3=0

Step 2

\(\displaystyle{x}^{{2}}+{y}^{{2}}−{19}{x}+{y}+{66}={0}\)....(1)

x−y−3=0..........................(2)

from eq(2) we get

x-y=3

x=y+3

now put the value of x in eq (1)

\(\displaystyle{\left({y}+{3}\right)}^{{2}}+{y}^{{2}}−{19}{\left({y}+{3}\right)}+{y}+{66}={0}\)

\(\displaystyle{y}^{{2}}+{6}{y}+{9}+{y}^{{2}}−{19}{y}−{57}+{y}+{66}={0}\)

\(\displaystyle{2}{y}^{{2}}−{12}{y}+{75}−{57}={0}\)

\(\displaystyle{2}{y}^{{2}}−{12}{y}+{18}={0}\)

\(\displaystyle{y}^{{2}}−{6}{y}+{9}={0}\)

\(\displaystyle{\left({y}-{3}\right)}^{{2}}={0}\)

y=3,3

Step 3

now as x=y+3

x=3+3

x=6

The solutions are (3,6) , (3,6)

we have the given two equations

\(\displaystyle{x}^{{2}}+{y}^{{2}}−{19}{x}+{y}+{66}={0}\)

x−y−3=0

Step 2

\(\displaystyle{x}^{{2}}+{y}^{{2}}−{19}{x}+{y}+{66}={0}\)....(1)

x−y−3=0..........................(2)

from eq(2) we get

x-y=3

x=y+3

now put the value of x in eq (1)

\(\displaystyle{\left({y}+{3}\right)}^{{2}}+{y}^{{2}}−{19}{\left({y}+{3}\right)}+{y}+{66}={0}\)

\(\displaystyle{y}^{{2}}+{6}{y}+{9}+{y}^{{2}}−{19}{y}−{57}+{y}+{66}={0}\)

\(\displaystyle{2}{y}^{{2}}−{12}{y}+{75}−{57}={0}\)

\(\displaystyle{2}{y}^{{2}}−{12}{y}+{18}={0}\)

\(\displaystyle{y}^{{2}}−{6}{y}+{9}={0}\)

\(\displaystyle{\left({y}-{3}\right)}^{{2}}={0}\)

y=3,3

Step 3

now as x=y+3

x=3+3

x=6

The solutions are (3,6) , (3,6)