 Hello and welcome to the session. In this session we shall discuss relationship between independent and dependent variables. As we have discussed earlier, in our system independent variables are taken as inputs which can be given specific values freely and dependent variables are the variables that are determined by other variables in the expression or we can say their values depend on the values of the independent variables. Now we shall discuss the relationship between independent and dependent variables. Now we see the relationship between two or more variables can be additive or multiplicative. Common relationships between two or more quantities are more than, less than and so on which are used for additive relationships and twice, twice, two times, three times etc. are used for multiplicative relationships. When we use these relationships between variables we obtain linear equation. Let us see some of the relationships. Suppose x and y are two variables, consider the following statements y is 5 more than x, y is 25 less than 50 times x and y is 7 times x. In all the three statements we see that y is the dependent variable and x is the independent variable. These can be written as first y is 5 more than x that is it can be written as y is equal to 5 plus x. Next is y is 25 less than 50 times x that is y is equal to 25 less than that is minus of 25, 50 times x that is 50 x. So we have y is equal to 50x minus 25 and y is 7 times x that is y is equal to 7x. Thus we have seen how to use these relationships between variables to obtain a linear equation. Now we shall learn that a relationship between two variables is said to be linear if they can be represented in a straight line on a graph. In other words we can say that linear equation is an equation for a straight line. Let us take an example. Suppose that there is a certain number of students entering in a classroom of grade 6. The number of girls is 2 more than 3rd times the number of boys in the classroom. We shall see how this situation can be expressed as an equation and how it can be represented on the graph. Here let the number of girls be x and the number of boys be y. Suppose boys enter in the classroom that is if y increases then the number of girls entering the classroom will be 4 times that is number of girls x is 4 times the number of boys that is y and when there are no boys in the class then there are already 2 girls in the classroom. So the equation so formed will be the number of girls x is equal to 4 times the number of boys that is y plus 2. So x is equal to 4y plus 2 is the required equation. Here we shall first identify the independent and the dependent variables. In this example the number of boys depend on the number of boys in the class so we can say that the number of girls x is the dependent variable and the number of boys y is the independent variable. Now we shall draw the graph of this equation and before drawing this linear equation on the graph we shall draw a table to find the value of both the variables so that we can obtain ordered pairs to plot on the graph. We draw a table with two columns one for the value of x and the other for the value of y. On the top in the first column we write x is equal to 4y plus 2 and in the second column we write y. Now below the second column we write three different values we want to give to the independent variable y and we shall find the corresponding values of x in the first column. And here we take 2 1 0 as the values of y as x is equal to 4 into y plus 2 so we have in the first column we put the value of y as 2 and we get x is equal to 4 into 2 plus 2 which is equal to 4 plus 2. 4 into 2 is 8 and 8 plus 2 is 10. So we have the value of x as 10. Now again if we put the value of y as 1 in the equation x is equal to 4 into y plus 2 we get x is equal to 4 into 1 plus 2 which is equal to 4 into 1 is 4 plus 2 is given by 6 so we have the value of x as 6. And again by putting 0 as the value of y in the equation x is equal to 4 into y plus 2 we get x is equal to 4 into 0 plus 2 which is equal to 2 so we get the value of x as 2. So we obtained the values of x as 10, 6 and 2 and here we obtained the ordered pairs as 10, 2, 6, 1 and 2, 0. Now let us see how to plot these points on the graph. Take x axis and y axis perpendicular to each other and mark positive numbers 1, 2, 3, 4, 5, 6, 7 and so on above the x axis and on the right of the y axis. Similarly mark minus 1, minus 2, minus 3, minus 4 and so on below the x axis and on the left side of the y axis as shown. Now we shall mark these points on the graph. Let us first take the point 10, 2. Now take 10 on the x axis and 2 on the y axis. Draw perpendicular from these points so that they meet at a certain point on the graph and this point is marked as 10, 2. Similarly we shall mark the point 6, 1 on the graph. We take 6 on the x axis and 1 on the y axis and we draw perpendicular from these points so that they meet at a certain point on the graph and we mark this point as 6, 1. And similarly we shall mark the point 2, 0. Now we join these 3 points on the graph and we obtain a straight line that represents the equation x is equal to 4y plus 2. Thus we can say it is a linear equation. Thus we have seen that a linear equation is an algebraic equation in which each term is a constant the product of constant with a variable. We also notice that a linear equation can have 1 or more variables. For example let us consider the equation x plus 3 is equal to 5. This equation consists of only one variable so it is called a linear equation in one variable. Now consider another example that is the equation y is equal to 2x plus 3. In this linear equation we have two variables y and x and 3 is a constant and this is called a linear equation in two variables. Here y is dependent on the values of x. So y is the dependent variable and x is the independent variable and this equation can be written in the statement form y is equal to 3 more than 2 times x. So here y and x have a combination of both additive and multiplicative relationship. This completes our session. Hope you enjoyed this session.