8.1. Addition and Subtraction of Algebraic Expressions

In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x^2, 4xy + 7 etc.

Algebraic expressions are combinations of numbers, variables, and operations (like addition or subtraction). They can represent mathematical relationships and can be as simple as a single term like x or as complex as 4xy + 7. The key is that they involve at least one variable.

Imagine you have a box that can hold a varying number of items. You can express the number of items in the box as x, which can change. If you add some fixed items, like 3, the total can be expressed as x + 3, showing how the contents of the box can vary.

To add 7x^2 – 4x + 5 and 9x – 10, we do:

7x^2 – 4x + 5
+ 9x – 10

This results in 7x^2 + 5x – 5. Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown.

When adding algebraic expressions, it’s important to line up like terms vertically. Like terms are terms that have the same variable raised to the same power. For example, – 4x and 9x are like terms, and you can add their coefficients: – 4 + 9 = 5. You also add constant terms together, such as 5 and –10, resulting in –5.

Think of adding apples and oranges. If you have 7 apples (represented by 7x^2) and 5 apples (5) from another basket, and you later find 9 oranges (9x), you can express your total in terms of apples and oranges, combining only the apples appropriately.

Example 1: Add: 7xy + 5yz – 3zx, 4yz + 9zx – 4y, –2xy – 3zx + 5x. Solution:

7xy + 5yz –3zx
+ 4yz + 9zx – 4y
+ –2xy –3zx + 5x

Results in 5xy + 9yz + 3zx + 5x – 4y.

In this example, we add three expressions by aligning like terms. For 7xy, 5yz, and –2xy, we group terms with the same variables. The result is a cohesive expression that represents the total combining all three individual expressions.

Imagine you have three different boxes of fruit. One has 7 of one type, another has 5, and the third –2 (meaning you owe 2). You can combine these to find out how much fruit you have in total, just like the expressions are combined to find one total expression.

To subtract 5x^2 – 4y^2 + 6y – 3 from 7x^2 – 4xy + 8y^2 + 5x – 3y, we write:

7x^2 – 4xy + 8y^2 + 5x – 3y
– (5x^2 – 4y^2 + 6y – 3)

This step allows us to see clearly which terms are being subtracted.

Subtraction works similarly to addition, but you need to also change the signs of each term in the expression you are subtracting. Essentially, subtracting can be viewed as adding a negative. For example, subtracting 5x^2 translates to adding –5x^2.

If you have $7 in one pocket and owe someone$5 (which is akin to subtracting it), you can think of it as putting your total into one equation where you start with positive funds and then deducting your debt to find your final balance.

Important: Subtracting a number is the same as adding its additive inverse. Thus, subtracting –3 is the same as adding +3. Similarly, subtracting 6y is like adding –6y; the same applies for –4y^2.

This concept emphasizes that subtraction can often be simplified by changing the operation to addition. Understanding the additive inverse helps to clarify points of confusion, particularly when encountering negative numbers.

Imagine you owe your friend $3 (subtracting$3 from your balance). When considering your money, it’s helpful to think of this as not just losing, but also as an incoming balance of $3 that you will need to account back to them from your future earnings.

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