 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that solve the system of equations graphically x plus 2y is equal to 10 and 3x minus y is equal to 2. Now we know that if the lines intersect that is if two lines L1 and L2 are intersecting lines then there exists unique solution and if the lines are parallel that is line L1 and line L2 are parallel then there exists no solution if the lines coincide that is if we have two lines L1 and L2 which are coinciding then there exists infinitely many solutions with this key idea let us proceed to the solution we have given equations that is x plus 2y is equal to 10 and 3x minus y is equal to 2 now we write these equations in slope intercept form and we know that any equation in slope intercept form can be written as y is equal to mx plus c and here we take the first equation that is x plus 2y is equal to 10 and it can be written as 2y is equal to minus of x plus 10 dividing both sides by 2 we get y is equal to minus of 1 by 2x plus 10 divided by 2 that is 5 and thus we have written the first equation in slope intercept form now we take the second equation that is 3x minus y is equal to 2 and it can be written as y is equal to 3x minus 2 and again we have got this equation in slope intercept form and now we will solve these equations that is y is equal to minus of 1 by 2x plus 5 and y is equal to 3x minus 2 graphically first we need to draw the graph of each linear equation so we take equation 1 that is y is equal to minus 1 by 2x plus 5 now let us draw an input output table for this equation for this we input values of x to find corresponding values of y now we will draw a table and we first put x as minus 2 we know that y is given by minus 1 by 2 into x plus 5 and then we put x as minus 2 we get y is equal to minus 1 by 2 into minus 2 plus 5 and we get y is equal to minus 1 into minus 1 that is 1 plus 5 so we get the value of y as 6 for x is equal to minus 2 we get the value of y as 6 similarly if we put the value of x as 0 that is if x is equal to 0 then we get y is equal to minus 1 by 2 into 0 plus 5 which implies that y is equal to 5 so for x is equal to 0 we get the value of y as 5 similarly when we take the value of x as 2 we get the value of y as 4 also if x is equal to 4 then we get the value of y as 3 now we shall plot these points on the coordinate plane and we get the ordered pairs as minus 2, 6, 0, 5, 2, 4 and 4, 3 now we plot these ordered pairs on the coordinate plane and the first ordered pair we have is minus 2, 6 so we take minus 2 on x axis and 6 on y axis and we put a dot here next we plot 0, 5 on the coordinate plane and we take 0 on x axis and 5 on y axis and we put a dot here and this is the point having coordinates 0, 5 and similarly we shall plot 2, 4 and 4, 3 on the coordinate plane and this is the point having coordinates 2, 4 and here is the point having coordinates 4, 3 now we join these points and this is the required graph of the equation y is equal to minus 1 by 2 x plus 5 now we will draw an input output table for the equation y is equal to 3x minus 2 and here also we will put values of x to get the corresponding values of y we know that y is equal to 3x minus 2 so when we put the value of x as 0 that is x is equal to 0 we get the value of y as 3 into 0 minus 2 that is 0 minus 2 which is equal to minus 2 so for x is equal to 0 y is equal to minus 2 now if we put the value of x as 1 we get the value of y as 3 into 1 minus 2 which is equal to 3 minus 2 that is 1 so for x is equal to 1 we have y as 1 similarly if we put the value of x as 2 we get the corresponding value of y as 3 into 2 minus 2 which is equal to 6 minus 2 that is 4 so for x is equal to 2 y is equal to 4 now we got our ordered pairs as 0 minus 2 1 1 and 2 4 and now we shall plot these ordered pairs on the same coordinate plane now our first ordered pair is 0 minus 2 so we take 0 on x axis and minus 2 on y axis and we put a dot here and we get this point which has coordinates 0 minus 2 now we shall plot the point which has coordinates 1 1 on the coordinate plane so we take 1 on x axis and 1 on y axis and we get the following point which has coordinates 1 1 and similarly we shall plot the point 2 4 and here is the required point 2 4 now we join all these points and hence we get the graph of the equation that is y is equal to 3 x minus 2 now we have got the two lines that is y is equal to minus 1 by 2 x plus 5 and y is equal to 3 x minus 2 graphically and we see that the two lines are intersecting each other in the coordinate plane and the point of intersection is this which has coordinates 2 4 and fun the key idea we know that if the lines are intersecting then there exists a unique solution that is if we have two lines L1 and L2 which are intersecting then there will be a unique solution and here as we see that the two lines are intersecting each other so there exists a unique solution and which is given by the ordered pair 2 4 so we can say that there is a unique solution given by the ordered pair 2 4 that is x is equal to 2 and y is equal to 4 which is the required answer now we can also check our answers by putting the value of x as 2 and y as 4 in the given equations that is we have an equation y is equal to 3 x minus 2 now if we put the values of x and y in this equation we get 4 is equal to 3 into 2 minus 2 which implies that 4 is equal to 6 minus 2 that is 4 is equal to 4 that is the left hand side is equal to the right hand side we can also verify the results with the other equation if we put the value of x as 2 and y as 4 in this equation we get 4 is equal to minus 1 by 2 into 2 plus 5 which implies that 4 is equal to minus 1 plus 5 that is 4 is equal to 4 and here also we have left hand side is equal to right hand side since the point 2 4 is satisfying both the equations so we can say that the point having coordinates 2 4 is the required solution of the given system of equations this completes our session hope you enjoyed this session