 Hello and welcome to the session. In this session, we will derive equations of straight lines of the form y is equal to mx and y is equal to mx plus b. Now, in our earlier sessions, we have discussed about slope of a line and proportional relationships. Now, let us recall the concept of proportional relationship and hence derive linear equation y is equal to mx. Now, two quantities are proportional if a ratio of the two quantities remains constant. That means if x and y are two quantities then the ratio y upon x is equal to k. This means y and x are proportional and here k is called constant of proportionality. For example, Sam covers the distance of 60 kilometers per hour at the following table in which distance of 60 kilometers is covered in one hour, then distance of 180 kilometers is covered in 3 hours and 30 kilometers is covered in 3 hours. Now, here d is the distance covered and h is the number of hours. Now, let us find the ratio of distance to hours to be 120 upon 2 which is again equal to 60. Then the next one 3 which is equal to 60 and the last ratio is which is again is equal to or equal to a constant. It means all these quantities are in proportion and here 60 is the constant of proportionality that is where k is equal to 60. d upon h is equal to relationship between d and h. Now, replacing d by y we get y is equal to, now let us find the rate of change which means we will find the slope which is acceptable to find the slope. So, consider these two. So, slope is equal to this is equal to 60 upon 1 which is the graph of a proportional relationship origin and it is a straight line. So, whenever we graph a proportional relationship between two variables x and y, we get an n dash we have the equation y is equal to mx and the relationship as d is equal to then we are getting a relationship between y and x. We get a straight line which passes through the origin. It means here y is proportional to x and the slope of this line which is passing through the origin. Relationships which have graphs are called linear relationships. The equation of straight line origin and slope m is y is equal to m. Now, let us discuss another linear equation which is y is equal to mx plus b also called slope intercept form statement. The pizza costs 7 dollars. Now here we have drawn a table in which in the first row we will write the number of pizzas and in the second row we will find the total cost of pizza which will include cost of pizza is 1. Then the total cost including cost of delivery will be 7 dollars which is equal to if number of pizzas are 2 then total cost will be 7 which is equal to we get the total cost for 3 pizzas and 4 pizzas. Now we have taken number of pizzas as x and total cost as y and now we will find the ratio y upon x the ratio y upon x then we see that each ratio is different. It means from the table for rate of change 2 points 1 minus now for these 2 points rate of change will be equal to change in y that is 6 to the minus 1 change in x that is 2 minus 1 upon 1 which is 7. Now let us take the points 1 and 9 now rate of change will be equal to 23 minus 9 whole upon which is equal to 14 upon 2 which is equal to and now consider the 2 points 1 and 9 and 4 30. Now for these 2 points rate of change is equal to 9 whole upon 4 minus 1 which is equal to 1 3 which is equal to we are getting the rate of it means there is a constant rate of change. Now from the table you can see when x is equal to 1 then y is equal to different values of x in this relationship we get the corresponding values of the relationship between x and y in equation form is y is equal to this amount being added will be a linear relationship it means its graph will be a straight line is a non-preferential relationship so it will not pass through the origin is given by constant rate of change so here slope is equal to 7. Now see when we put x is equal to 0 in this equation then we get y is equal to 7 into 0 plus 2 which is equal to 2 it means this line will pass through the ordered pair 0 2 this means it is a point where this line 1 plus 2 is the y intercept now where we can see when we plot the graph of the equation y is equal to 7 x plus 2 then we get a straight line and this straight line passes through the point 0 2 and you can see that at this equation of the straight line having slope 7 and y intercept 2 is given down y is equal to 7 x plus plus and y intercept b i is equal to an x plus we have derived is equal to n x and y is equal to if you all have enjoyed the session