Basic mathematical operations: combination of similar terms
In math, you have four main operations: addition, subtraction, multiplication, and division. Since subtraction is the inverse of addition, multiplication is repeated addition, and division is the inverse of multiplication, you will see that the other three operations follow indirectly from addition. In this sense, there really is a binary operation in mathematics: addition. Binary operation refers to the use of a mathematical operator, such as addition, on two numbers or variables, such as x + y. Since we see how important addition is now, we must fully understand one of the most important tasks in all of mathematics: that of combining like terms.
similar terms are expressions that involve the same combination of variables and their respective exponents but different numerical coefficients. The coefficients, if you remember, are the numbers in front of the variable. To put this in simple terms, like terms are like apples and apples, oranges and oranges. Examples of like terms are 4x and 2xgold 3 years and 9 years. To take the abstraction out of this whole thing, the student should keep in mind that whenever the expressions are similar regardless of the coefficients, the terms can be added or subtracted. Thus 3xy and 4xy are like terms and can be combined to give 7xy. Take away the coefficients 3 and 4, and what’s left? xy.
Many times a student will not be able to arrive at the final answer to an algebra problem because at some point the like terms were not combined correctly. In more complicated math problems, the expressions can get a bit more complicated. However, if you take into account that similar terms are similar “animals”, so to speak, then, like animals, they can safely mate. If the terms are not similar, he can never combine them. The results are always disastrous. What usually helps students is to bring them out of abstraction and face to face with hard facts: If two algebraic expressions, after removing the numbers in front of them, look the same, then they are like terms and can be added and subtracted. Note that we are only talking about the two operations of addition and subtraction, as these are the two operations that require the terms to be similar before being combined. Multiplication and division do not have this requirement.
Let’s look at some examples to make this perfectly clear and see where some potential problems can arise. Let’s do the examples below.
1) 3x + 18x
2) 8xyw – 3xyw + xyw
3) 3x^2 – x^2 + 6x
The first example can be thought of as 3 x and 18 x. Think of the actual plastic-shaped letter in a child’s game. Obviously, you have either 21 x or 21x as the answer.
The second example gives an indication of when students might start to have problems. The moment more than one letter or variable is introduced, students are quickly intimidated. not be If you remove the coefficients of each of the terms, you see that they are all x and w terms. The last term has a coefficient of 1, which is understood. Combining, we have 6xyw.
The third example introduces an expression with exponents. Remember that the exponent, or power, only tells us how many times to use the number as a factor when multiplying by itself. So x^2 tells us to multiply x by itself, ie x^2 = x*x. If you remove the coefficients in this example, you’ll see that you have 2 x^2 terms and one x term. So you can only combine the terms x^2. The answer becomes 2x^2 + 6x. Please note that terms that cannot be combined simply remain as they are.
The information here should make you a master of combining like terms as this is actually a very easy but extremely important task. If you follow the precepts laid out here, you should have no more difficulty simplifying basic algebraic expressions.