Step 1

The system of equations is given by,

\(\displaystyle{x}^{{2}}+{\left({y}-{9}\right)}^{{2}}={49}\) (1)

\(\displaystyle{x}^{{2}}-{7}{y}=-{14}\) (2)

Step 2

Solve the system of equations.

\(\displaystyle{\left({7}{y}-{14}\right)}+{\left({y}-{9}\right)}^{{2}}={49}\)

\(\displaystyle{7}{y}-{14}+{y}^{{2}}-{18}{y}+{81}-{49}={0}\)

\(\displaystyle{y}^{{2}}-{11}{y}+{18}={0}\)

\(\displaystyle{y}^{{2}}-{9}{y}-{2}{y}+{18}={0}\)

\(\displaystyle{y}{\left({y}-{9}\right)}-{2}{\left({y}-{9}\right)}={0}\)

\(\displaystyle{\left({y}-{2}\right)}{\left({y}-{9}\right)}={0}\)

\(\displaystyle{y}={2},{9}\)

Step 3

The solution of the system of equations is computed as follows.

For y=2 , the value of x is \(\displaystyle{x}^{{2}}={7}{\left({2}\right)}-{14}\)

\(\displaystyle{x}^{{2}}={14}-{14}\)

\(\displaystyle{x}^{{2}}={0}\)

x=0

For y=9, the value of x is \(\displaystyle{x}^{{2}}={7}{\left({9}\right)}-{14}\)

\(\displaystyle{x}^{{2}}={63}-{14}\)

\(\displaystyle{x}^{{2}}={49}\)

\(\displaystyle{x}=\pm{7}\)

The solution is \(
(0,2),(-7,9),(7,9)\)