 Hello and welcome to the session. In this session, we will discuss how to find solution of two simultaneous linear equations. Graphically, we know that any linear equation is of the type y is equal to mx plus b where m is the slope and b represents the y-intercept. Now suppose if we are given any two linear equations that is y is equal to ax plus b and y is equal to cx plus d and we have to find the solution of these two equations that is we have to find that ordered pair or pairs which will satisfy both the equations and that ordered pair will be the solution of the system of equations and now we are going to learn how to find the solution of system of equations graphically and before moving on we shall learn about different methods to graph a line and the first method says we can graph a line using slope intercept form that is if we given equations r of the form 2x plus 3y is equal to 4 and 3x plus 6y is equal to 9 then we can write these equations in slope intercept form that is y is equal to mx plus b that is the equation 2x plus 3y is equal to 4 can be written as 3y is equal to 4 minus 2x or it can also be written as 3y is equal to minus of 2x plus 4 now dividing by 3 on both the sides we get y is equal to minus of 2 by 3x plus 4 by 3 and hence we get y is equal to minus of 2 by 3x plus 4 by 3 which is of the form y is equal to mx plus b and similarly we have the other equation that is 3x plus 6y is equal to 9 which can be written as 6y is equal to minus of 3x plus 9 now dividing by 6 on both the sides we get y is equal to minus of 3 by 6 into x plus 9 by 6 which implies that y is equal to minus of 1 by 2x plus 3 by 2 and hence we get the equation that is y is equal to minus of 1 by 2x plus 3 by 2 which is again of the form y is equal to mx plus b and now we can graph these equations using an input output table that is we put the values of x and we will get corresponding values of y and we have to take at least 3 points for x and we choose values for x according to convenience x can be given any value like 0, 1, minus 1, 2, minus 2 and so on and we put these values of x to find corresponding values of y and then we plot the pairs on the coordinate plane and join them to get a straight line and hence in this way we get the required graph for any linear equation and we can also use x and y intercepts to graph a straight line that is for x intercept we put y is equal to 0 in the equation and for y intercept we put x is equal to 0 in the equation and then we plot the two points on the coordinate plane and join them and in this way we will get the required straight line and now we will discuss steps to find the solution of two simultaneous linear equations graphically and first we shall draw the graph of the two equations on the same coordinate plane and we can graph the equations by any of the above mentioned methods. Next we will check whether the lines are intersecting parallel or coincident and then we will write the solution from graph and now we are going to discuss types of solutions and their representation on graph. Now there are three types of solutions first is the unique solution second no solution and third infinitely many solutions and we will discuss all of these one by one and first is unique solution or only one solution that is there exists only one ordered pair which satisfies both the equations graphically we can say that the lines drawn will intersect at only one point and that point is the required solution suppose we have to solve two equations that is x plus y is equal to 6 and 2x minus y is equal to 6 and we can write these equations in slope intercept form that is we can write these equations in the form of y is equal to mx plus b where m is the slope and b is the y-intercept so we write this first equation as y is equal to minus of x plus x and the second equation can be written as y is equal to 2x minus 6 and now we have the two equations in slope intercept form now for different values of x we can obtain corresponding values of y and thus for different values of x we have obtained the corresponding values of y for equation 1 that is y is equal to minus of x plus 6 if we put the value of x as 0 we get the corresponding value of y as 6 for x is equal to 1 y is equal to 5 for x is equal to 2 y is equal to 4 and for x is equal to 3 we get the value of y as 3 similarly for the second equation that is y is equal to 2x minus 6 for different values of x we have obtained the corresponding values of y that is for x is equal to 0 y is equal to minus 6 for x is equal to 1 we get the value of y as minus 4 for x is equal to 2 y is equal to minus 2 and for x is equal to 4 we get the value of y as 2 and now by using these two tables we shall graph these equations on the coordinate plane now for equation 1 that is y is equal to minus of x plus 6 we have our first order there as 0 6 so we take 0 on x axis and 6 on y axis and we put a dot here our next order there is 1 5 so we take 1 on x axis and 5 on y axis and we put a dot here and similarly we shall plot the next two ordered pairs that is 2 4 and 3 3 now we join all these points and this is the required graph for the equation y is equal to minus of x plus 6 and similarly we shall draw the graph for the second equation that is y is equal to 2x minus 6 by plotting these ordered pairs on the graph and the ordered pairs are 0 minus 6 1 minus 4 2 minus 2 and 4 2 and hence we have plotted these points on the graph now we join all the points and here is the required graph of the equation y is equal to 2x minus 6 and here we see that the two lines intersect each other at one point which has coordinates 4 2 so there exists a unique solution given by the coordinates 4 2 that is x is equal to 4 and y is equal to 2 we can also verify this answer by putting x is equal to 4 and y is equal to 2 in the given set of equations so in the first equation that is y is equal to minus of x plus 6 if we put the value of x as 4 and the value of y as 2 we get 2 is equal to minus of 4 plus 6 which implies that 2 is equal to 2 therefore the left hand side is equal to the right hand side also in the second equation that is y is equal to 2x minus 6 if we put the values of x and y as 4 and 2 respectively we get 2 is equal to 2 into 4 minus 6 that is 2 is equal to 2 into 4 that is 8 minus 6 which is equal to 2 so here also the left hand side is equal to the right hand side so we say that x is equal to 4 and y is equal to 2 satisfies both the equations thus the point which has coordinates 4 2 is the required solution next we are going to discuss no solution there are system of equations which may not have any solution it means there does not exist any other pair which satisfies the given system of equations graphically we can say that the two lines have same slope and thus they are parallel to each other now let us consider the system of equations y is equal to 2x plus 1 and y is equal to 2x minus 3 here we can see that the slope of both the lines is given by 2 so graphically they must be parallel and now we draw graph of both the equations and here we have drawn the graph for the first equation that is y is equal to 2x plus 1 and this is the graph of the second equation that is y is equal to 2x minus 3 and now if we see the two lines we notice that the two lines are parallel because they have same slope that is 2 so there exists no solution and next we have infinitely many solutions that is there exists a system of equations which can have infinite solutions and the graph of the two equations will be coincident it means the straight lines overlap each other or coincide with each other and the solution is the set of all the points of other pairs line on the line and we'll get same straight line for both linear equations and the lines will have same slope and same intercepts if we consider the system of equations that is y is equal to 3x minus 2 and 2y is equal to 6x minus 4 we clearly see that the equation 2y is equal to 6x minus 4 and dividing by 2 on both the sides can be written as y is equal to 3x minus 2 it means both the equations are same so the graph will have one straight line and we get the following graph we see that the two lines coincide therefore this system of equations will have infinitely many solutions and thus these system of equations will have infinitely many infinitely many solutions and the solution is the set of all points of other pairs line on the line so in this session we have discussed how to find the solution of two simultaneous linear equations graphically that's all for this session hope you have enjoyed this session