 Hello and welcome to the session. In this session we discuss the following question that says, show graphically that the given system of equations has infinitely many solutions. x-2y equal to 5 and 3x-6y equal to 15. Now, we are given a system of equations and we need to show that this system of equations has infinitely many solutions. Let's see what is the condition for which the given system of equations has infinitely many solutions. For this, first consider a system of equations a1x plus b1y is equal to c1. Let this be equation 1. a2x plus b2y is equal to c2. Let this be equation 2. We consider a line l1 which represents the graph of equation 1 and line l2 represents the graph of equation 2. Now, if the lines l1 and l2 are coincident then we say the system has infinitely many solutions or you can say the system is consistent. This is the condition for the system to be consistent or to have infinitely many solutions. This is the key idea for this question. Now, we proceed with the solution. The given system of equations is x-2y equal to 5. Let this be equation 1. 3x-6y is equal to 15. Let this be equation 2. Now, we will draw the graph of both these equations. First consider the equation x-2y equal to 5. Now, we will find two points on the line which represents this equation. When we take x equal to 1 in this equation, we get 1-2y equal to 5. That is, minus 2y is equal to 5 minus 1 equal to 4. So, y is equal to 4 upon minus 2 which is equal to minus 2. So, for x equal to 1, we get y equal to minus 2. Now, when we take x equal to 5. So, we have 5 minus 2y equal to 5 which means minus 2y equal to 0 or we get y is equal to 0. So, when we take x equal to 5 in this equation, we get the value of y as 0. So, for the equation 1, we have got two points. Say a point a with coordinates 1-2 and a point b with coordinates 5-0. These two points lie on the line x-2y equal to 5. Let's plot these two points on the graph. First, we take the point a with coordinates 1-2. Now, here the x-coordinate is 1 and the y-coordinate is minus 2. So, from this point, we move two steps downwards along the y-axis since the y-coordinate is minus 2. So, this point is the point a with coordinates 1-2. The other point is b with coordinates 5-0. So, the x-coordinate is 5 and the y-coordinate is 0. So, this point would represent the point b with coordinates 5-0. Now, we join the points a and b. So, we have got this line a b which represents the equation x-2y equal to 5. Now, let's consider the other equation that is 3x-6y equal to 15. Now, we draw the graph for this equation. Now, we will put some values for x in the given equation and find the corresponding values of y. Like for x equal to 1, we get 3 into 1 minus 6y equal to 15. That is minus 6y is equal to 12 and from here we get y is equal to minus 2. And when we put x equal to 5 in the given equation, we get 3 into 5 minus 6y is equal to 15 which gives us minus 6y is equal to 0 that is y is equal to 0. So, for x equal to 5, the corresponding value of y thus obtained is 0. So, now we get 2 points say point c with coordinates 1-2 and a point d with coordinates 5-0 and these 2 points lie on the line 3x-6y is equal to 15. Now, we plot these 2 points c and d of the graph. First, consider the point c with coordinates 1-2. x-coordinate is 1, y-coordinate is minus 2. So, this point will represent the point c with coordinates 1-2. Then we have the point d with coordinates 5-0, x-coordinate is 5 and the y-coordinate is 0. So, this point represents the point d with coordinates 5-0. So, this line that is the line joining the point c and d which is the same as the line joining the point c and d represents the equation 3x minus 6y equal to 15. Thus, we have that cd represents the equation 3x minus 6y equal to 15. Now, from the graph as you can see that av and cdr coincident lines av and cd are coincident and we know that if lines L1, L2 represents the equations of a given system of linear equations and the 2 lines are coincident, then we say that the given system is consistent or it has infinitely many solutions. Now, as the lines av and cd representing the equations of the given system of linear equations are coincident that we say the given system of linear equations has infinitely many solutions. So, with this we complete the session. Hope you have understood the solution for this question.