Question

Solve the given system of equation
\[x + y + xy = 5\]
\[{x^2} + {y^2} + xy = 7\]

Answer 1

Hint: We are provided two system equations \[x + y + xy = 5\] and \[{x^2} + {y^2} + xy = 7\] .
We are going to simplify it using substitution and completing the square method.
The substitution method is the mathematical method to solve linear equations. In this method, we substitute the value of one variable from one equation in the other equation so as to solve the equation.
Completing The Square Method is one of the methods to find the roots of the given quadratic equation. In this method, we have to convert the given equation into a perfect square by dividing the coefficient of the x term and then adding it.

FORMULA USED: Use the given identity \[{a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2}\]

Complete step-by-step answer:
Let,
\[x + y + xy = 5\] ………. (I)
\[{x^2} + {y^2} + xy = 7\] ………(ii)
In algebra, we know that
\[{a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2}\]
Using this identity simplify equation (ii)
\[{x^2} + {y^2} + xy = 7\]
=\[{x^2} + {y^2} + 2xy - xy = 7\]
=\[{\left( {x + y} \right)^2} - xy = 7\] ……(iii) [\[{x^2} + {y^2} + 2xy = {\left( {x + y} \right)^2}\]]
Simplifying equation (i)
\[x + y + xy = 5\]
\[x + y = 5 - xy\] ……(iv)
Putting value of (iv) in iii
\[{\left( {5 - xy} \right)^2} - xy = 7\]
Using \[{a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2}\]
\[25 + {x^2}{y^2} - 10xy - xy = 7\]
Rearranging and simplifying constant term
\[{x^2}{y^2} - 11xy + 18 = 0\]
Simplify more through completing the square method
\[{x^2}{y^2} - 2 \times xy \times \dfrac{{11}}{2} + {\left( {\dfrac{{11}}{2}} \right)^2} - {\left( {\dfrac{{11}}{2}} \right)^2} + 18 = 0\] [\[{\left( {xy - \dfrac{{11}}{2}} \right)^2} = {x^2}{y^2} - 2 \times xy \times \dfrac{{11}}{2} + {\left( {\dfrac{{11}}{2}} \right)^2}\]]
\[{\left( {xy - \dfrac{{11}}{2}} \right)^2} - \dfrac{{121}}{4} + 18 = 0\]
\[{\left( {xy - \dfrac{{11}}{2}} \right)^2} = \dfrac{{49}}{4}\]
Moving square to right hand side
\[xy - \dfrac{{11}}{2} = \pm \sqrt {\dfrac{{49}}{4}} \]
\[xy - \dfrac{{11}}{2} = \pm \dfrac{7}{2}\]
CASE 1:
When \[xy - \dfrac{{11}}{2} = \dfrac{7}{2}\]
\[xy = \dfrac{7}{2} + \dfrac{{11}}{2}\]
\[xy\]\[ = \]\[\dfrac{{18}}{2}\]
\[xy = 9\] ……(v)
CASE2:
When \[xy - \dfrac{{11}}{2} = - \dfrac{7}{2}\]
\[xy = \dfrac{{11}}{2} - \dfrac{7}{2}\]
\[xy\]\[ = \]\[\dfrac{4}{2}\]
\[xy\]\[ = \]2 ……. (vi)
Hence, the solution of the given pair of equations will be every pair of x and y which satisfies any of equation (v) or (vi).
So, there will be infinite solutions for this equation.

Note: Remember to simplify the complex equation through any method like substitution, completing the square method. In case of infinitely many solutions \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\]
and the line will be a coincident line.
The linear equation has unique or infinite solutions.
The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-slope. If the two lines have the same y-intercept and the slope they coincide with each other.