 Hello and welcome to the session. In this session we are going to discuss how to apply the properties of operations and use them as strategies to add, subtract, factor and expand linear expressions with rational coefficients. Now consider a linear expression or we can say another bright linear expression that is 5x plus 2y plus 3z. This type of linear expression is divided into three parts that is variables, operations and coefficients variables such as x, y and z as used in this linear expression and the operations used in this linear expression are addition and multiplication numbers like 5, 2 and 3 that are used in this linear expression are called coefficients. The coefficients in a linear expression are rational that is integers, positive or negative fractions or decimals. Then these are called rational coefficients. Now let us learn how to apply properties of operations and work with rational coefficients and first we have associative property which says addition or multiplication in numbers or in algebra remains the same no matter how the numbers are grouped. For example we will verify 2 plus 3 plus 5 in brackets is equal to 2 plus 3 in brackets plus 5 here on the left hand side we have 2 plus 3 plus 5 in brackets. So we first simplify the terms in brackets and later add 2 to it so we get 2 plus which is equal to and similarly on the right hand side we have 2 plus 3 in brackets plus 5. So again we simplify the brackets first and then add 5 to it and therefore we get 5 plus 5 which is equal to 10 here both left hand side and right hand side are equal to 10 and same applies for multiplication that is 2 into 3 into 5 in brackets is equal to 2 into 3 in brackets into 5 and this is equal to 30 that is on the left hand side we have 2 into 3 into 5 in brackets. So here also we simplify the brackets first and we get 2 into 15 which is equal to 30 and similarly on the right hand side we have 2 into 3 in brackets into 5 so we get 6 into 5 that is simplify the brackets first which is equal to 30 as it rightly we can write associated property for addition as A plus B in brackets plus C is equal to A plus B plus C in brackets and for multiplication it is A into B in brackets into C is equal to A into B into C in brackets. Now we are going to discuss commutative property to fasten the buttons of your shirt it does not matter that you fasten the last button first or the first button last so the order in which you fasten your buttons does not matter that if you are eating something then you choose first and then swallow you cannot reverse this order that is you cannot swallow first and then chew later so in case of fastening the buttons of your shirt the order in which you fasten the buttons does not matter so it is commutative but the order in which you eat does matter so it is not commutative so we say that an operation commutative in the order of the numbers does not change the results that is for example 2 plus 3 is equal to 3 plus 2 that is equal to 5 that is if we add 3 2 2 or if we add 2 2 3 the results will always be 5 so we can say that it is commutative similarly we have 2 into 3 is equal to 3 into 2 that is 6 also in algebra we can write a plus b is equal to b plus a that is for addition and a into b is equal to b into a that is for multiplication now if the distributive property which involves both addition and multiplication suppose we have to solve 3 into n plus n in brackets where n and n are any numbers to multiply m and n by number 3 which is outside the brackets that is 3 into n plus n can be written as 3 into m plus 3 into n and using the same method we can solve 3 into 7 plus 2 and we get 3 into 7 plus 3 into 2 so solving any expression using the other rule is called distributive property and if we solve left hand side and right hand side separately we get the same value here on left hand side we have 3 into 7 plus 2 in brackets on simplifying the brackets first we get 3 into 7 plus 2 that is 9 which is equal to 3 into 9 that is 27 and on right hand side we have 3 into 7 plus 3 into 2 that is we get 3 into 7 is 21 plus 3 into 2 that is 6 21 plus 6 will be equal to 27 hence we see that both the results are equal to multiply a sum or difference by a number multiply each number in the sum or difference by the number outside the parenthesis that is if we have a into b plus c in brackets so we can write it as a into b that is a b plus a into c that is a c or if we have a into b minus c in brackets we can write it as a into b that is a b minus a into c that is a c now let us take a few examples and understand these strategies to add, subtract, multiply, divide, factor and expanding linear expressions now let us consider 5x plus 3x that is 5x can be written as x plus x plus x plus x that is adding x 5 times plus 3x can be written as x plus x plus x so here we get the total as apex for addition we add life times together where coefficients of x are 5 and 3 so we add 5 and 3 and we get so we say that 5x plus 3x is equal to apex similarly we have subtraction now consider the expression 2x minus 3x we can also write this expression as 2x plus or minus 3x by using the properties of integers that is we get 2x can be written as x plus x plus minus of 3x can be written as minus x minus x minus x and therefore we get x plus x plus of minus x is minus x again plus of minus x is minus x plus of minus x is minus x and therefore we are left with minus x now if we directly apply the properties of integers to the coefficients in the expression we get 2 minus 3 into x that is we subtract the two integers and the sign of the result would be the sign of the larger integer so 2 minus 3 will be equal to minus 1 into x and we can write it as minus x here the larger integer is 3 and it has a negative sign so our answer would be negative next we have multiplication consider 10x into 5x here we multiply the coefficients together and the variables together and therefore we get 10 into 5 into x into x which is equal to 10 into 5 is 50 into x into x that is x square so our answer is 50x square here we should note that we obtained a nonlinear expression as the power of x is 2 next we move on to the reason if you have the expression 10x upon 5x to simplify this expression we first divide the coefficients that is 10 is divided by 5 and we get 2 next we divide the variables that is x is divided by x and the answer is 1 and we get 2 into 1 that is 2 so we get 2 as the answer next we are going to discuss factors and expansion if we consider the expression that is 10 times x plus 2 on expansion we can write it as 10 into x 10 into 2 which is equal to 10x plus 20 also if we want to write 2x plus 10 in the factor form we can write 2x plus 10 as 2 into x plus 2 into 5 now taking to common from both the terms we get 2 into x plus 5 now to check whether 2x plus 10 is equal to 2 into x plus 5 in brackets now 2x plus 10 is equal to 2 into x plus 5 implies that 2x plus 10 upon 2 is equal to x plus 5 if we take the expression on the left hand side we have 2x plus 10 whole upon 2 and we can write it as 2x upon 2 plus 10 upon 2 which is equal to x plus 5 and it is equal to the expression on the right hand side is complete our session hope you enjoyed this session