 Hello and welcome to the session. In this session we are going to discuss how to solve and check the linear equations involving fractions and grouping symbols. We know that a linear equation in one variable is an equation of the type AX plus B is equal to C where the exponent of the variable is 1. To solve a linear equation means to find the values of X for which left hand side of the equation is equal to the right hand side of the equation. Now to solve an equation we write the given equation in the form variable is equal to constant by simplifying grouping symbols like parenthesis, brackets and braces and simplifying the algebraic expressions. Let us consider an example that is solve for X. We have to solve this equation for X and now we shall follow the following steps. Now our first step is to remove inner most grouping symbols. Here inner most grouping symbol is parenthesis. So first we open these round brackets and we get 5 into 3X minus 2 into 4 plus 3 into X that is 3X then plus 3 into minus 1 that is minus 3 the whole and this complete whole and this is equal to 5 into X that is 5X plus 5 into 6 that is 30. Now we open these square brackets inside the parenthesis and we get 5 into 3X now minus 2 into 4 is minus 8 minus 2 into 3X will be minus 6X then minus 2 into minus 3 is plus 6 the whole is equal to 5X plus 30. Now we remove the remaining grouping symbol that is parenthesis and we get 5 into 3X that is 15X 5 into minus 8 is minus 40 5 into minus 6X is minus 30X 5 into 6 is plus 30 and this is equal to 5X plus 30. Now in the next step we combine light terms here 15X and minus 30X are light terms also minus 40 and plus 30 are light terms. So we write 15X minus 30X minus 40 plus 30 is equal to 5X plus 30 and this implies that now 15X minus 30X will be minus 15X minus 40 plus 30 will be minus 10 and this is equal to 5X plus 30. Now we write this obtained equation in variable is equal to constant form and thus we write minus 15X minus 5X is equal to 30 plus 10 and here we get minus 20X is equal to 40. Now dividing both sides by minus 20 we get minus 20X upon minus 20 is equal to 40 upon minus 20 this implies that X is equal to minus 2. Now let us check our solution for this we put X is equal to minus 2 in the original equation here we get 5 into 3 into minus 2 minus 2 into 4 plus 3 into minus 2 minus 1 the whole this whole and then this complete whole and this is equal to 5 into minus 2 plus 6 the whole this implies that 5 into now 3 into minus 2 is minus 6 minus 2 into 4 plus 3 into minus 2 minus 1 that is minus 3 the whole and this complete whole and this is equal to 5 into minus 2 plus 6 is 4 this further implies that 5 into minus 6 minus 2 into 4 plus 3 into minus 3 that is minus 9 the whole and this complete whole is equal to 5 into 4 that is 20 this further implies that 5 into minus 6 minus 2 into now 4 minus 9 is minus 5 the whole is equal to 20 which further implies that 5 into minus 6 minus 2 into minus 5 is plus 10 the whole is equal to 20 that is 5 into now minus 6 plus 10 is 4 is equal to 20 that is 20 is equal to 20 so left hand side is equal to right hand side. And so we can say that X is equal to minus 2 is the solution of the given equation now we are going to discuss linear equations involving fractions and grouping symbols. To solve linear equations involving fractions and grouping symbols we will follow following steps the first step is to remove the parenthesis if any then we multiply each term of the equation by the least common denominator of all fractions appearing in the equation then we combine light terms on each side of the equation and write the equation in variable is equal to constant form and then we will solve it for X let us consider the following example here we have to solve this equation for X in the first step we will remove the grouping symbol parenthesis and we get 1 by 2 into X that is 1 by 2 X 1 by 2 into 3 that is plus 3 by 2 minus 1 by 4 X is equal to 1 by 2 into X that is 1 by 2 X plus 1 by 2 into 2 that is 2 by 2 which is equal to 1 now here we can see that the least common denominator of all the fractions here is equal to 4 so we multiply each term of this equation by 4 so that we can get rid of the fractions we get 4 into 1 by 2 X plus 4 into 3 by 2 minus 4 into 1 by 4 X is equal to 4 into 1 by 2 X plus 4 into 1 that is 4 this implies that 2 into X is 2X plus 2 into 3 is 6 minus minus 1 into X will be minus X and this is equal to 2X plus 4 this further implies that 2X minus X will be equal to X plus 6 is equal to 2X plus 4 now bring variable terms on one side of the equation and constant terms on the other side here we get X minus 2X is equal to 4 minus 6 this implies that minus X is equal to minus 2 now dividing both these sides by minus 1 we get X is equal to 2 now we will check whether X is equal to 2 is the solution of the equation or not now we put X is equal to 2 in the original equation that is 1 by 2 into X plus 3 the whole minus 1 by 4 X is equal to 1 by 2 into X plus 2 the whole and therefore we get 1 by 2 into 2 plus 3 the whole minus 1 by 4 into 2 is equal to 1 by 2 into 2 plus 2 the whole this implies that 1 by 2 into 2 plus 3 that is 5 minus 1 by 2 is equal to 1 by 2 into 2 plus 2 that is 4 this further implies that 5 by 2 minus 1 by 2 is equal to 2 and this further implies that 5 minus 1 whole upon 2 is equal to 2 that is equal to 4 by 2 is equal to 2 and this implies that 2 is equal to 2 so left hand side of the equation is equal to right hand side of the equation thus X is equal to 2 is the solution of the given equation this way we can solve linear equations involving grouping symbols and fractions this completes our session hope you enjoyed this session