Add the following expressions using both the horizontal method and column method:
$2 a b - 3 a^{2} b + 7 a b^{2}, 4 a b^{2} + 5 a^{2} b - 3 a b$ and $7 a^{2} b - 10 a b^{2} + 3 a b$

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Solution

Step 1: Adding the algebraic expressions by the horizontal method
Horizontal method: Taking care of signs, write all expressions in a horizontal line and group like terms together. Now, add these like terms and get the required sum.
We have $2 a b - 3 a^{2} b + 7 a b^{2}, 4 a b^{2} + 5 a^{2} b - 3 a b$ and $7 a^{2} b - 10 a b^{2} + 3 a b$
Now,
$(2 a b - 3 a^{2} b + 7 a b^{2}) + (4 a b^{2} + 5 a^{2} b - 3 a b) + (7 a^{2} b - 10 a b^{2} + 3 a b) = 2 a b - 3 a^{2} b + 7 a b^{2} + 4 a b^{2} + 5 a^{2} b - 3 a b + 7 a^{2} b - 10 a b^{2} + 3 a b = 2 a b - 3 a b + 3 a b - 3 a^{2} b + 5 a^{2} b + 7 a^{2} b + 7 a b^{2} + 4 a b^{2} - 10 a b^{2} = (2 - 3 + 3) a b + (- 3 + 5 + 7) a^{2} b + (7 + 4 - 10) a b^{2} = (2) a b + (9) a^{2} b + (1) a b^{2} = 2 a b + 9 a^{2} b + a b^{2}$
Step 2: Adding the algebraic expressions by the column method
Column method: Write each expression in separate lines such that their like terms lie in the same vertical column. Taking care of signs, add each column.
$2 a b - 3 a^{2} b + 7 a b^{2} - 3 a b + 5 a^{2} b + 4 a b^{2} + 3 a b + 7 a^{2} b - 10 a b^{2}2 a b + 9 a^{2} b + a b^{2}_$
Hence, the required sum is $2 a b + 9 a^{2} b + a b^{2}$

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