Addition and Subtraction of Algebraic Expressions

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Introduction to Addition and Subtraction of Algebraic Expressions

Algebraic expressions can have several variables like \({a, \;b,\;and\;c\;or\;x,\;y\;and\;z}\). Each variable can again be raised to different powers, like \({a^2,\,x^3,\, b^2y,\, or\, c^3z}\) or any other combination. Each such term can again be multiplied by different constants like 2, 6, or even 25.

For us to be able to add or subtract algebraic expressions, the most important thing to do is to align similar terms together and then apply the operator. This brings us to the concept of like terms.

The Big Idea: What are like terms?

Take a look at the example below and remember your concepts of coefficients, constants and variables. The thing to keep in mind while trying to add or subtract algebraic expressions is that the rules for adding or subtracting expressions translate perfectly to algebra from numbers as long as we are adding terms with the same variable or like terms.

This brings us to the definition of like terms. Two different algebraic expressions are called like terms when they have the same set of variables raised to the same set of powers. They may or may not have the same constant (or numerical co-efficient). So, \({3x^2y}\) and \({6x^2y}\) are like terms, but \({3x^2y}\) and \({4x^2y}\)4 are not like terms.

Some common pitfalls

The most common error is to mix up similar looking expressions. For example, \({xy^2}\) looks very similar to \({x^2y}.\) If you are not careful, the terms \({3xy^2}\) and \({4x^2y}\) might be considered as like terms, and the solution would be wrong.

The commutative properties would apply here as well. So \({3xy}\) and \({6yx}\) are like terms even though they might look different.

The need for like terms

Why do you need to club all the like terms together? Well, to understand this, think back to what variables represent. Each variable represents a different parameter of an expression. At the end of the day, an algebraic expression is simply a statement converted to the language of mathematics. Just like in english, we can only compare likes, the same holds true for algebraic expressions.

As you can see, terms with the same variables, represent the same thing in an algebraic expression which allows us to compare them.

Addition and Subtraction

Let us understand the process with the help of an example or two. Let us consider the following two expressions \({E_{1}} \) and \({E_{2}} \).

\({E_{1}= -2x + 3yx\; – 2x^2y + 5y \;– 10xy^2}\)

\({E_{2}= x^2y \;– 9xy^2 + 4x\; – 3y + 2xy + 10}\)

The first thing you will notice here is that the variables are not in the same order in both expressions. Another thing to notice is that the \({+3yx}\) and the \({+2xy}\) in both the respective expressions are actually like terms. And finally, the second expression \({E_{2}} \) has an additional constant term (+10) which doesn’t have a corresponding constant in the first expression \({E_{1}} \). Keeping these three points in point, let us write the two expressions with all like terms aligned.

\({E_{1}= -2x + 5y + 3xy – 2x^2y – 10xy^2}\)

\({E_{2}= +4x \;– 3y + 2xy- x^2y\; – 9xy^2 + 10}\)

Once this is done, then we simply need to add the coefficients of the like terms. Therefore,

\({E_{1}+E_{2}= (-2x + 4x) + (3xy + 2xy) + (-2x^2y - x^2y) + (-10xy^2 - 9xy^2) + 10}\)

     \({= 2x + 5xy -3x^2y -19xy^2 + 10}\)

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