Question

How do you simplify $8a + b - 3a + 4b$?

Answer 1

Hint: Sum of the polynomials is the addition of the polynomials with their sign observation. Summation always gives us the resultant amount of the function with respect to their sign convention. Sign conventions imply that if the polynomial function carries a negative sign with it then, the addition of the polynomial with another polynomial will result in the actual subtraction of the polynomial instead of addition. For example, when $x, - x,5x$ are added together then the results will be $x - x + 5x = 5x$ instead of $x + x + 5x = 7x$.
Before starting to solve the question, it should be also observed carefully whether the polynomials have the same degree (order) or not. Degree is the sum of the highest power of the function. For example, the functions $2x,2xy,2{x^2}y$ have the degree as 1, 2 and 3 respectively. In the questions all the functions are of degree 1.
In this question, sign convention is the foremost thing to be taken care of while adding the polynomials.

Complete step by step solution:
As all the polynomials have the same degree i.e., 1, then the summation of the numbers will also be of degree 1.
Summation of $8a,b, - 3a,4b$ is calculated as:
$S = 8a + b - 3a + 4b$
Reorder the elements in such a way that the same variables group together.
$ \Rightarrow S = 8a - 3a + b + 4b$
Now, take $a$ common from the first two elements and $b$ from the last two elements.
$ \Rightarrow S = \left( {8 - 3} \right)a + \left( {1 + 4} \right)b$
Add $8, - 3$ and $1,4$ as shown below
$ \Rightarrow S = 5a + 5b$
Now, take $5$ common from the elements.
$ \Rightarrow S = 5\left( {a + b} \right)$
Hence, $8a + b - 3a + 4b = 5\left( {a + b} \right)$.

Note:
Alternatively, this question can also be solved by adding all the polynomials which have a positive sign and adding all the polynomials which have a negative sign and then, at last subtracting the polynomial having a negative sign with the polynomial having the positive sign.