 Hello and welcome to the session. In this session we will discuss a question which says that solve the given system of equations algebraically and the equations are given as x square plus y square is equal to 20 and y is equal to x plus 2. Now before starting the solution of this question we should know a result. And that is method to solve the linear quadratic system algebraically. Now in the first step in the given system of equations since y is the independent variable in the quadratic equation so substitute for y then write the new formed quadratic equation in standard form and such is equal to 0 then solve this quadratic equation then in the next step substitute the obtained values in the linear equation to find the corresponding values of y then write the solutions as ordered pairs of numbers then check the ordered pairs in each of the original equations So this is the method to solve the linear quadratic system algebraically and this result will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Now in this question we are given the system of equations or let this be equation number 1 and this be equation number 2. Now substitute the value of y from equation 2 in equation 1. So substituting the value of y that is y is equal to x plus 2 in equation 1 we have x square plus x plus 2 whole square is equal to 20. This implies x square plus now x plus 2 whole square is x square plus 2 square that is 4 plus 2 into 2 into x that is is equal to 20. This implies 2x square plus 4x plus 4 minus 20 is equal to 0 and this implies 2x square plus 4x now plus 4 minus 20 is minus 16 is equal to 0. Now dividing this whole equation by 2 we have x square plus 2x minus 8 is equal to 0. Now this is a quadratic equation in x so let us solve it by factorization. Now by splitting the middle term this is x square plus 4x minus 2x minus 8 is equal to 0. Now this implies now from these 2 terms taking x common it will be x into x plus 4 the whole and from these 2 terms taking minus 2 common it will be minus 2 into x plus 4 the whole is equal to 0. This implies x plus 4 the whole into x minus 2 the whole is equal to 0. Now this implies x is equal to minus 4 and x is equal to 2 so we have 2 values of x that is x is equal to minus 4 and x is equal to 2. Now according to the key idea when we have solved the quadratic equation then we will substitute the obtained values in the linear equation to find the corresponding values of y. Now in the given system the second equation is the linear equation so we will put x is equal to minus 4 and x is equal to 2 in this linear equation 1 by 1 to obtain corresponding values of y. First of all let us put x is equal to minus 4 in equation 2 so we have y is equal to minus 4 plus 2 which implies y is equal to minus 2. Now let us put x is equal to 2 in equation 2 and we have y is equal to 2 plus 2 which implies y is equal to 4 so for x is equal to minus 4 y is equal to minus 2 and for x is equal to 2 y is equal to 4. So we have of the given system of equations first is the ordered pair minus 4 minus 2 to 4. Now we check whether these points satisfy both the equations or not. First of all let us take the ordered pair minus 4 minus 2 now we will put x is equal to minus 4 and y is equal to minus 2 in equation 1 and we have minus 4 whole square plus of minus 2 whole square is equal to 20 which implies 16 plus 4 is equal to 20 which gives 20 is equal to 20 which is true. Now let us check it for the second equation so here we will put x is equal to minus 4 and y is equal to minus 2 in equation 2 and we have minus 2 is equal to minus 4 plus 2 which implies minus 2 is equal to minus 2 which is also true. The ordered pair minus 4 minus 2 is the solution of the given system of equations. Now let us take the second ordered pair that is the ordered pair 2 4 first of all let us put x is equal to 2 and y is equal to 4 in equation 1 and we have 2 square plus whole square is equal to 20 which implies 4 plus 16 is equal to 20 and this gives 20 is equal to 20 which is true. Then let us check this for the second equation now for the second equation by putting x is equal to 2 y is equal to 4 we have 4 is equal to 2 plus 2 which implies 4 is equal to 4 which is also true so the ordered pair 2 4 is the solution of the given system of equations thus the solution of the given quadratic linear system is the ordered pair minus 4 and the ordered pair 2 4. So that is all for this session hope you all have enjoyed the session.