What are the Basics of Algebra?

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

An algebraic expression is a collection of variable numbers and constants that combine with one, two, or more operations like addition (plus), subtraction, multiplication, division, or a rational exponent. An algebraic expression can also be said to include a group of one, two, or more terms, where a term consists of factors — either number (digit) or variable(s) or both together, with or without an exponent.

A variable is generally denoted by x or y or some other letter whose value is not known or can be anything.

A constant is a number by itself.

A coefficient is a number accompanying a variable (multiplication).

An exponent tells us how many times a variable or a number is multiplied by itself.

Interesting Fact

The word algebra is derived from the Arabic word “Al-Jabr”, and is taken from the title of a book written by a mathematician from Persia, Muhammed ibn Musa Al-Khwarizmi, who is also acknowledged as the forefather of algebra.

How to Subtract?

Subtraction in algebra is both related to and different from how basic subtraction is usually done with numbers. The difference being that only similar numbers or like terms can be subtracted while the rest or unlike terms can only be included in the algebraic expression and cannot be combined.

This can be easily explained by using a corollary of desserts. Just like one cannot take away a sundae from a cake or a brownie as all three are essentially very different from each other, similarly only like terms can be added or subtracted. If you are really hungry you can of course eat all three, but that doesn’t have anything to do with algebra!

Subtraction Strategies

Compensating

This is a strategy that is used the most.

So, let’s take a look at what it would be like if we were doing 16 – 9 using the compensating strategy. First, we would be subtracting 16 – 10 which would be 6. But after that, there is a tricky part. We need to visualize 16 blocks and take away 10. This needs compensation because we did something we weren’t supposed to do. Now we need to come back and compensate for it. We subtracted 1 more than what was supposed to, so we need to give that 1 back to compensate for what we did, i.e., we need to add 1 to the difference, 6, to obtain that 16-9=7.

Give and Take (Constant Difference)

There are two ways to think about subtraction. There’s the way that is known as take-away, where we’re viewing the problem as it is. The other way to view subtraction is actually what the answer to a subtraction problem is called, i.e., the difference. What you’re really trying to figure out in subtraction is the difference between the subtrahend and the minuend. That is the simple way to say it is the difference between two numbers.  If it was 9 and 16, again, 16 – 9, we could shift this up to 10. We might want to make that a 10. But then we have to shift the 16 to make that difference stay the same. We shift up one both ways, give one and then give another one to the other side and that keeps the distance the same. In this case, the problem 16-9 becomes 17-10, which gives the same result as 7.

Adding Up (Finding the Distance) 

A lot of ways in which subtraction can be done involve adding up, because it is a whole lot easier than subtracting.

Now if we go back to our 16-9 problem, again, it’s about finding that difference. We could start at 9 and probably hop up one to get to the 10. And then do a big hop of 6 to reach 16. Since we hopped for 1 and 6 places, so the difference of 7. Here the answer is in those hops. We have to add those up to figure out what is the answer for the subtraction problem.  

Decomposing 

Another one of the strategies is decomposing. It’s just like decomposing in science. It’s breaking it down into smaller chunks.

The hard part though is to have to keep track of what has been subtracted. And that’s why we sometimes will think about the hundreds, then the tens, then the ones. But we don’t have to subtract, like, the 9 all at once. We can see that we could chunk the nine into something friendlier like a 3 and then a 6. Now what this would look like when we are doing 16 – 9 is well, 9 is not very friendly here. But hey, a 6 is. So that would get us to a 10, and then we could take away the remaining 3 to get to the 7. The idea is that we are breaking it down into friendlier pieces to subtract.

Place Value/Like Values

Another popular strategy is when we want to break by place value. That is a place value or like value strategy. The idea is that we are looking at trying to subtract like values. Like subtracting ones from ones, tens from tens, hundreds from hundreds. This progresses through all that we are going to be doing in subtraction. We want to be able to see that we have to have like values. We can’t just add halves with fourths. We need to have fourths with fourths.

For example, let’s see how 289 is subtracted from 546.

So, what we have is 543 which is really a 500, a 40, and a 6. And we need to take away a 200, we need to take away an 80, and then take away a nine from that. We start here at 6 and can’t take away a 9, so we come over and regroup a 10 and so on.

It’s a 10 and a 6 and we need to subtract a 9. Well, we don’t want to take the 9 from the 6, because there’s not enough there. So we break it apart and then find ways to be able to match these up and subtract it.

Traditional Algorithm

It is called traditional because it’s the traditional way that we all learned. And it’s an algorithm because you can program a computer to do it. It is solved the same way every single time. It’s the same steps that you go through to solve the problem. We start in our ones. When there are not enough ones, we regroup or borrow. The reason we call it regrouping is that we’re just taking some of the numbers and regrouping them into something that makes a little bit more sense.

Subtraction of Algebraic Expressions

Two ways to practice math problems on the subtraction of algebraic expressions are:

Horizontal Method

In this method, all expressions are written in a horizontal line, the signs of the terms in the minuend are changed, and then the terms are regrouped to collect all the groups of like terms and then added.

Column Method

In this method, each expression is written in a separate row such that they are regrouped one below the other in a column. The signs of the terms in the minuend are changed, and then the addition of numbers is done column-wise. 

Rules of Algebra

There are a few rules that have to be followed while doing operations on algebraic expressions. These are:

• Commutative law of addition
• Commutative law of multiplication
• Associative law of addition
• Associative law of multiplication
• Distributive laws

The use of algebra is so widespread, that a lot of times one doesn’t even realize that it is being used in synonym for everyday calculations and other languages. Suppose that you have to calculate the amount of money you need to carry with you for a shopping trip to a nearby grocery store. If you need a dozen eggs (e), 2 quarts of milk (m), 3 loaves of bread (b), and 5 ounces of cheese (c), the expression for your shopping expedition(S) will be:

S = 12e + 2m + 3b + 5c

Now if the shopkeeper knows you and gives you a discount of 10% on 1 ounce of cheese with every purchase of a loaf of bread, you will get a discount of 0.1c each for the 3 loaves of bread, and the equation changes to:

S = (12e + 2m + 3b + 5c) – (3x0.1c)

Assuming the prices for one egg, one quart of milk, one loaf of bread, and an ounce of cheese to be $0.25, $3, $5, and $4 respectively, the total price for your shopping will be:

S = 12x0.25 + 2x3 + 3x5 + 5x4 – 3x0.1x4

S = 3 + 6 + 15 + 20 – 1.2

S = $ 42.80

Algebraic equations can be used in many situations of daily life where there are variables and constants involved in complex proportionalities to arrive at finite results. Simply put, algebra is everywhere, even if it is used unconsciously most of the time.

Context and Applications

The concept of algebraic subtraction is greatly used in K-12 examinations, as well as in graduate, and postgraduate studies, especially for:

• B.A Mathematics
• B.Sc. Mathematics
• Masters in Mathematics

Common Mistakes

• One of the most common mistakes made during the subtraction of arithmetic expressions is the use of incorrect operators in erroneous form while removing brackets or solving for them. When opening brackets, if there is a subtraction sign before it, then the signs of all the terms will need to be flipped.
• Double-check all the variables and their exponents carefully before the inversion and subsequent addition of like terms.
• The use of parentheses helps greatly in avoiding calculation mistakes in algebra.

Related Concepts

• Addition
• Multiplication
• Integers