Question

Add the following algebraic expression:
$\dfrac{11}{2}xy+\dfrac{12}{5}y+\dfrac{13}{7}x,-\dfrac{11}{2}y-\dfrac{12}{5}x-\dfrac{13}{7}xy$

Answer 1

Hint: Start by bringing together the terms in the addition with the same variable parts together. Then take the variable part common and make pairs of two terms and add the constant terms. For adding the constant terms take the LCM of the denominators and add accordingly.

Complete step-by-step answer:
Let us start the solution to the above question by representing the addition in expression form. We will just use an additional sign between the two expressions to show the addition.
$\dfrac{11}{2}xy+\dfrac{12}{5}y+\dfrac{13}{7}x+\left( -\dfrac{11}{2}y-\dfrac{12}{5}x-\dfrac{13}{7}xy \right)$
Now we will open the expression keeping in mind the signs of the terms.
$\dfrac{11}{2}xy+\dfrac{12}{5}y+\dfrac{13}{7}x-\dfrac{11}{2}y-\dfrac{12}{5}x-\dfrac{13}{7}xy$
Now we will arrange such that all the terms with the same variables are together.
$\dfrac{11}{2}xy-\dfrac{13}{7}xy+\dfrac{12}{5}y-\dfrac{11}{2}y+\dfrac{13}{7}x-\dfrac{12}{5}x$
Now we will take the variable parts common and add the constant parts. On doing so, we get
$xy\left( \dfrac{11}{2}-\dfrac{13}{7} \right)+y\left( \dfrac{12}{5}-\dfrac{11}{2} \right)+x\left( \dfrac{13}{7}-\dfrac{12}{5} \right)$
Now we will take the LCM of the denominators. To determine the LCM of the numbers, express the number in terms of the product of its prime factors and multiply all the prime factors the maximum number of times they occur in either number. As all the denominators are primes, the LCM would be their multiplications.
$xy\left( \dfrac{77-26}{14} \right)+y\left( \dfrac{24-55}{10} \right)+x\left( \dfrac{65-84}{35} \right)$
$=xy\left( \dfrac{51}{14} \right)+y\left( \dfrac{-31}{10} \right)+x\left( \dfrac{-19}{35} \right)$
$=\dfrac{51}{14}xy-\dfrac{31}{10}y-\dfrac{19}{35}x$
Hence, the answer to the above question is $\dfrac{51}{14}xy-\dfrac{31}{10}y-\dfrac{19}{35}x$ .


Note:Be very careful about the signs while opening the brackets as the general mistakes made by the students include $xy\left( \dfrac{51}{14} \right)+y\left( \dfrac{-31}{10} \right)=\dfrac{51}{14}xy+\dfrac{31}{10}y$ . Also, remember that the operations like addition and subtraction can only be done with the terms with the same variable part and are not possible with terms having different variables multiplied with the constants, the mistake that a student generally makes in this is $\dfrac{11}{2}xy+\dfrac{12}{5}y=y\left( \dfrac{11}{2}+\dfrac{12}{5} \right)$ , so be careful that which part have been taken common.