 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that write a linear function that relates y to x from the given table and we are given a table in which x values are given as minus 2, 0, 2 and 4 and we corresponding y values are given as 1, 0, minus 1 and minus 2. Now we know that in a linear relationship the rate of change is constant. With this key idea let us proceed to the solution. Now we are given an input output table of x and y values and we have to find the linear relationship between x and y. In the table we are given when x is minus 2, y is 1, when x is 0, y is also 0, when x is 2, y is minus 1 and when x is 4, y is minus 2. Now we will find change in the values of x and y, change in x is given by 0 minus 2 and is equal to 0 plus 2 that is 2. Also we have 2 minus 0 that is 2 and 4 minus 2 that is also equal to 2 and here we see that change in x is constant and is given by 2 that is if we add 2 to the previous value of x we will get the next value that is minus 2 plus 2 is 0, 0 plus 2 is 2 and 2 plus 2 is 4. Now we will see change in y and it is given by 0 minus 1 that is equal to minus 1 minus 1 minus 0 that is equal to minus 1 and minus 2 minus of minus 1 and is equal to minus 1. Now here again we notice that change in y is constant and is given by minus 1 that is if we add minus 1 to the previous value of y we will get the next value that is 1 plus of minus 1 is 0, 0 plus of minus 1 gives minus 1 and minus 1 plus of minus 1 gives minus 2 and from the key idea we know that in a linear relationship the rate of change is constant and here we see that the change in x and change in y is constant so we can now find the linear relationship between x and y. Now to find the linear relationship between x and y we shall first find the rate of change and rate of change is given by change in y upon change in x and change in y is given by minus 1 and change in x is given by 2 so rate of change is equal to minus 1 upon 2 and this gives us the slope of the line that is the value of m is equal to minus 1 upon 2. Now here from the table we see when the value of x is 0 y is also 0 and thus we say that the line will pass through the origin that is 0, 0 we know that equation of a line passing through the origin with slope m is given by y is equal to mx and here the value of m is minus 1 by 2 so we can write y is equal to minus 1 by 2 x thus y is equal to minus 1 by 2 x if the required equation that gives the linear relationship between x and y this completes our session. Hope you enjoyed this session.