# Solve the following systems of equations. x^2+y^2−19x+y+66=0 x−y−3=0

Question
Equations
Solve the following systems of equations.
$$\displaystyle{x}^{{2}}+{y}^{{2}}−{19}{x}+{y}+{66}={0}$$
x−y−3=0

2021-01-17
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)

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