The rules of addition of like terms are:
Rule I: When all the terms are positive, add their coefficients, also the variables and power of the like terms remains the same.
1. 5xy, 4xy and xy.
Here, 5xy, 4xy and xy are like terms
The sum of the coefficients = 5 + 4 + 1 [xy means 1xy]
Therefore, 5xy + 4xy + xy = 10xy
To add two or more like terms, we add the numerical coefficients of the given terms and form another like term with the sum obtained as the numerical coefficient of the resulting term.
2. 5x + 4x + 2y + 3y
Here, 2x and 3x are like terms
and also 6y and 5y are like terms
5x + 4x = 9x
2y + 3y = 5y
Therefore, the answer is 9x + 5y
Rule II: When all the terms are negative, add their coefficient without
considering their negative signs and then prefix the minus sign (-) to
1. -3ab, -5ab and -ab
Without considering the negative signs, the coefficients of the given terms are 3, 5 and 1 respectively; and 3 + 5 + 1 = 9.
Therefore, addition of -3ab, -5ab and –ab = -9ab
i.e. (-3ab) + (-5ab) + (-ab) = -9ab
2. -5x + (-4x) + (-2y) + (-3y)
Here, -5x and -4x are like terms
and also -2y and -5y are like terms
(-5x) + (-4x) = -9x
(-2y) + (-3y) = -5y
Therefore, the answer is -9x - 5y.
Rule III: When all the terms are not of same sign. The same rule as that for the addition of integers should be applied.
1. Addition of 21m and –9m
= 21m + (-9m)
= 21m - 9m
= m(21 - 9)
2. 9xy – 4xy – 5xy + 7xy – xy
= 5xy – 5xy + 7xy – xy
= 0 + 7xy – xy, [since, 5xy – 5xy = 0]