 Hello and welcome to this session. In this session, we shall discuss the concept of linear graphs. Before we discuss about the linear graphs, let's first discuss about the Cartesian system. Whenever we need to describe the position of any object, we can do this by giving the distance of that object from two reference lines. One would be the horizontal reference line which is the x-axis, then the other is the vertical reference line which is the y-axis. And the point of intersection of the x-axis and the y-axis is called the origin. This is the horizontal reference line which is O x, this is the x-axis and this O y is the vertical reference line which is y-axis and this point O is the point of intersection of the two axis. This point O is the origin. Now the position of any point is represented by a pair of numbers such as AB. This pair AB is said to be the ordered pair. Now since this ordered pair represents the position of any point, so we can say that AB represents the coordinates of the point. The first number of this ordered pair which is A is E x-coordinate at CISA and the second number which is B is to y-coordinate the ordinate. Suppose we have this point, so point A with coordinates AB. Now this x-coordinate of the point A that is A represents the perpendicular distance of the point A from the y-axis and the y-coordinate which is B represents the perpendicular distance of the point A from the x-axis. Now the axis can be extended to create four quadrants and when we count anticlockwise we refer it as first quadrant, second quadrant, third quadrant and this is the fourth quadrant. For the first quadrant the coordinates of a point would be plus plus that is the x-coordinate and the y-coordinate both would be positive. In the second coordinate x-coordinate negative y-coordinate positive in the third quadrant x-coordinate negative y-coordinate negative in the fourth quadrant x-coordinate positive and y-coordinate negative. Now in general we represent the point by the coordinates AB any point on the x-axis would be represented as A0 that is to y-coordinate in this case would be 0 or you can say the ordinate would be 0 and any point on the y-axis would be represented as 0B that is the axis R or the x-coordinate is 0. Let us now discuss the graphing of linear equations. A linear equation, a linear equation whose graph is a straight line. An equation of the form AX plus BY plus pulled to 0 where we have ADC are the real numbers and here A and B both are not equal to 0. This is a linear equation in two variables and the two variables here are X and Y. Now we will follow certain steps to graph a linear equation in two variables. First of all we write the given equation showing one variable in terms of the other. So if we are given an equation of this kind we can express X in terms of Y or Y in terms of X then in the next step we find at least values for these variables. Like for the variables X and Y we will find three sets of values. The next we draw the x-axis and the y-axis taking three plots. The three points step two we draw a line passing through these points. This line represents the graph of the given linear equation in two variables. Let us now graph the equation X plus Y equal to 6. As according to the first step we need to write the given equation in the form such that it shows one variable in terms of the other variable. So as the given equation is X plus Y equal to 6. So here we can express so Y in terms of the variable X as Y is equal to 6 minus. According to the next step we will find at least three sets of values for the variables X and Y. So we now make a table with the values of X and Y. Now when we take X as 0 in this equation we get the value of Y as 6. For X as 1 the value of Y would be 5. For X as 2 the value of Y would be 4. So we have got three ordered pairs say point A with coordinates 0, 6, B with coordinates 1, 5, C with coordinates 2, 4. We draw the x-axis and the y-axis taking a suitable scale. So we have drawn this x-axis and the y-axis. Next we plot the three points which we have obtained in this table that is these three points A, B and C. First we consider the point A with coordinates 0, 6 that is X coordinate is 0, Y coordinate is 6. So as the Y coordinate is 6 and the X coordinate is 0. So from the 0 we move 6 units above the x-axis. So we reach at this point. So this is the point A which has coordinates 0, 6. Next consider the point B with coordinates 1, 5 as the X coordinate is 1. So one unit to the right from the origin. So we reach at this point and as the Y coordinate is 5. So from here we will move 5 units above the x-axis. So we reach at this point. This is the point B with coordinates 1, 5. Next is the point C with coordinates 2, 4. The unit is 2. So we move 2 units to the right from the origin. We reach at this point and as the Y coordinate is 4. So we move 4 units above from the x-axis. That is from this point which gives us this point and this is the point C with coordinates. Now as we have plotted all the 3 points that we have obtained. Next we draw a line which passes through these points. So this line that we have obtained by joining the 3 points is the line X plus Y equal to 6. Now as we can see all the 3 points lie in a straight line. So this shows that we have not made any error while drawing this graph. Of the 3 points do not lie on the line. Then we can say that we have made an error in graphing the equation. So this is how we can graph any equation in 2 variables. Now the graph of a first degree equation in only one variable is either the x-axis, the y-axis or the x-axis. We can also say that the graph of y equal to 0 is the x-axis. Then the graph of x equal to 0 is y-axis. Then the graph of y equal to a is the line a units from it. And the graph of equal to b is the line equal to y-axis at the distance of b units from it. Now as you can see that the points a, b and c lie on the graph of the equation. So this means that these points would satisfy the given equation. Next we discuss solving a pair of simultaneous equations graphically. Solving a pair of simultaneous equations means that we find the points where the graph of the given equations cross each other. And that point is the point of intersection of the lines which will give us the solution of the simultaneous equations. So solving the pair of simultaneous equations graphically would involve two steps. We draw the graphs of two equations that is the simultaneous equations in the same coordinate plane. And then we find the intersection. The section would give us the solution of the simultaneous equations. Let us now solve the simultaneous equations x plus y equal to 6 and x minus y equal to 4. For this we will first draw the graphs of the two equations in the same coordinate plane. You know that you have already drawn the graph of the equation x plus y equal to 6. Next we consider the other equation. x minus y equal to 4, x plus y in terms of x. So y is equal to x minus 4. So when we take x as 4 in this equation we get y as 0. Taking x as 5 we get y as 1. x as 3 we get y as minus 1. So thus we obtain 3 points. So points D with coordinates 4, 0, E with coordinates 5, 1, F with coordinates 3, minus 1. Let us now plot these three points. First we have the point D with coordinates 4, 0. This is the point D with coordinates 4, 0. Next we have a point E with coordinates 5, 1. This is the point E with coordinates 5, 1 and a point F with coordinates 3, minus 1. So this is the point F with coordinates 3, minus 1. Now we will join these three points that is E, D and F to get the graph of the equation x minus y equal to 4. This line joining the points E, D and F is the graph of the equation x minus y equal to 4. This point E is the point of intersection of the two lines, two equations E with coordinates 5, 1. And therefore we say x equal to 5, y equal to 1 is the solution of the given simultaneous equations. So this is how we solve a pair of simultaneous equations graphically. Now when the points in a plane lie on a straight line, we get a graph which is a straight line. This is called a linear graph. So we say when in a plane lie on a straight line, we get a linear graph. So this completes the session. Hope you have understood the concept of linear graphs.