[MUSIC PLAYING] Let's look at our objectives for today. We will start with an introduction to when we combine like terms. We will then define terms and like terms. We'll look at how to combine like terms. We'll look at some common mistakes or errors made when combining like terms. And finally, we'll do some examples combining like terms.
So let's start by looking at when we combine like terms. Solving an equation for a single variable is not always a single step process. Oftentimes, complex expressions appear on either side of an equation and it requires several steps in order to isolate the variable. So simplifying the expression by combining like terms before attempting to isolate the variable can save time and prevent confusion.
Now let's look at the definitions of a term and like terms. A term is a collection of numbers, variables, and powers combined through multiplication. Here's an example of an expression with three terms. We have 6x squared plus 3x plus 2. We refer to terms by their variable and power or exponent.
The first term, 6x squared, is called an x squared term. We call the 6 in front of the x squared a coefficient. The 2 is the exponent or power of the term. The second term, 3x, is called an x term. The coefficient of this term is 3. Here the variable x has no written exponent, so the exponent is 1. The last term, 2, is a constant term. Terms without a variable are called constant terms because the value of that term will be the same no matter what the variable is.
So now that we know what terms are, let's look at what like terms are. Like terms have the same variables, with each variable having the same exponent. In our expression, there are no like terms, because in each term, the variable x has a different exponent. However, in the expression 6x squared plus 3x plus 5x plus 2, 3x and 5x are like terms. And we can simplify the expression by combining these two terms. So let's see how we combine like terms.
Combining like terms is a way to simplify expressions before solving the equation using inverse operations. Terms may only be added or subtracted if they are like terms. Like terms are combined by adding or subtracting their coefficients and leaving the variable and exponent the same. Again, a coefficient is the number that appears in front of a variable and acts as a factor or multiplier. Coefficients can be any real number, and terms without a written coefficient have an implied coefficient of 1.
So now let's look at some common mistakes or errors that should be avoided when combining like terms. One common mistake is changing the exponent when combining like terms. For example, 4x plus 3x does not equal 7x squared, because the exponent does not change. Instead, 4x plus 3x means we are adding 4 x's and 3 more x's, which would give us a total of 7 x's. So 4x plus 3x equals 7x.
Another common mistake is combining coefficients of terms that are not like terms. For example, 2x plus 2 does not equal 4x. Even though both terms have a 2, they are not like terms because their variable and variable exponent are not the same. So 2x and 2 cannot be combined or simplified, again, because they are not like terms.
Let's do some examples combining like terms. When combining like terms, it's useful to remember the commutative property for addition, which tells us that a plus b is equal to or the same as b plus a. In other words, we can add real numbers in any order. This property is useful when simplifying expressions involving like terms, because it allows the terms to be rearranged when we combine like terms using addition and subtraction.
Here's our first example. We have the expression 4x squared plus x plus 9 plus 3x. In this expression, we have two like terms, x and 3x. We know that the term x has a coefficient of 1 that we just don't write. So to combine the terms, we add the coefficients 1 and 3. 1x plus 3x is 4x, so our expression becomes 4x squared plus 4x plus 9.
Here's our second example. 2x to the third plus 3 minus 2x squared plus 11 plus 7x squared minus 4. So we first see that we have two x squared terms. The subtraction sign in front of the 2x squared makes the term a negative 2x squared. This can be combined with 7x squared. Adding our coefficients, negative 2 and 7, gives us 5. So negative 2x squared plus 7x squared simplifies to 5x squared.
We then see that we have three constant terms, 3, 11, and negative 4. We can combine these three terms. 3 plus 11 minus 4, or 3 plus 11 plus negative 4, gives us 10. The remaining term, 2x to the third, cannot be combined with any other terms in the expression, so it is left unchanged. And our final expression is 2x to the third plus 5x squared plus 10.
Here's our last example. We have 1/5 x to the third plus 3/4 x minus 1/2 x plus 1/5 x to the third. We have two x cubed, or x to the third powered terms which can be combined. We need to add their coefficients 1/5 and 1/5. Since their denominators are the same, we can simply add the numerators. So 1/5 plus 1/5 is 2/5. So 1/5 x cubed plus 1/5 x cubed becomes 2/5 x cubed.
We can then combine our two x terms. We want to add their coefficients, 3/4 and negative 1/2. Here the denominators are not the same, so we need to find a common denominator. The least common denominator would be 4, so we leave the fraction 3/4 unchanged, and multiply the fraction 1/2 by 2 in the denominator and the numerator, which will give us negative 2/4.
Now the denominators are the same, so we have 3/4 plus negative 2/4. We can add our numerators, which will give us 1/4, so the two x terms simplify to 1/4 x. Our simplified expression is 2/5 x cubed plus 1/4 x.
Let's look at our important points from today. Make sure you get these in your notes so you can refer to them later. Simplifying expressions by combining like terms before attempting to isolate the variable can save time and prevent confusion. A terms is a collection of numbers, variables, and powers combined through multiplication. We refer it terms by their variable and power or exponent. Terms without a variable are called constant terms, because the value of that term will be the same no matter what the variable is.
Terms may only be added or subtracted if they are like terms. And finally, like terms are combined by adding or subtracting their coefficients and leaving the variable and exponent the same. So I hope that these important points and examples helped you understand a little bit more about combining like terms. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.
