Combinação de termos iguais

Combining like terms is sort of like cleaning up a messy sock drawer. I don't even like to open my sock drawer because every time I do, I'm met with polka dot socks, white socks, tall socks, short socks, and socks that I'm not even sure are mine.

Complex equations can look like a messy sock drawer, with variables and like terms scattered all over the place. The approach to manage these complex equations is the same one you'd take to clean up that messy drawer: match up terms that go together (we call these "like terms" - think of them like matching socks).

It feels darn near impossible to solve an equation that's not simplified—just like your chances of finding a particular sock in that messy drawer would be pretty much zero. So, let's take a look at a few examples on how we can "clean up" these equations.

This neat trick will be immensely helpful when tackling large, complex equations with a bunch of different variables and constants. Practicing this skill is essential to helping you solve for $x$!

Take a look at this equation. Pretty messy, right? $$3x + 2 + 4(y - 1) = 5y - 8x - 7$$

To start, let's define like terms. Like terms are different terms that share the same variable and power. In our equation here, $3x$ and $-8x$ are like terms since they both have an $x$. Notice how we include the variable and its coefficient (the number in front of it). Similarly, any terms containing $y$ are like terms with each other. And finally, all the terms without $x$ or $y$ in them are also considered like terms (constants).

Now that we have that established, let's get to combining like terms! The first thing we need to do is get rid of those pesky parentheses. Recall that things with parentheses need to be handled first according to the Order of Operations (PEMDAS). Use the distributive property to get rid of the parentheses. $$\begin{align} 3x + 2 + 4(y - 1) &= 5y - 8x - 7\\ 3x + 2 + 4y - 4 &= 5y - 8x - 7 \end{align}$$

Now that we've got our equation down to a form with only numbers, variables, and simple operations like addition and subtraction, we can start combining like terms.

For this example, it doesn't matter which side of the equation you put them on or whether they're together. Recall that it is most common to format your equation, though, in a way that gives you as many positive values as possible.

$$11x + 5 = y$$

Again, this is just one way of writing the simplified equation. We chose this approach because it kept all of our values positive. Here we can see that combining like terms helped us to see the equation more clearly.

Let's look at another equation, this time with three variables ($x$, $y$, and $z$).

$$2z -3 + \frac{x}{2} + 4y = x - 12 + 6(z + 1)$$

That's a pretty messy equation with fractions, but it's totally manageable. Let's follow the same approach we took before: handle parentheses and operations like multiplication and division, and then combine like terms.

We're going to opt to change our $\frac{x}{2}$ term into $0.5x$. Think of $\frac{x}{2}$ as the same as $(\frac{1}{2})x$. Since $\frac{1}{2} = 0.5$ it's completely fine for us to rewrite our $\frac{1}{2}$ as $0.5$. This instantly gets rid of the pesky division operation in the equation. Now, we've got an equation with a decimal.

$$2z -3 + 0.5x + 4y = x - 12 + 6(z + 1)$$

Now, handle the parentheses via the distributive property.

$$\begin{align} 2z -3 + 0.5x + 4y &= x - 12 + 6(z + 1)\\ 2z -3 + 0.5x + 4y &= x - 12 + 6z + 6 \end{align}$$

Now we're ready to combine our like terms.

$$4y = 0.5x - 3 + 4z$$

Just like before, it's up to you how you arrange these terms. We chose this arrangement because it kept most of our terms positive.

Also, you are free to change the decimal back to a fraction in your simplified equation, or you can leave it as it is. It's the same information either way!

Let's do one more example. Solve for $x$ by grouping the like terms in the equation: $$2x+5x-3-6+9-12=6x+4+1+5+7-2+7x-3$$

It's a good idea to double check your answer for the $x$ value when you're solving these types of equations.

(Video) Like Terms

by mathman1024

Here's a video by mathman1024 explaining like terms. Like terms are terms that have the same variable part to them and matching exponents.

$\large{\text{Summary}}$

For example: $3x$ and $4x$ are like terms because they both have $x$ as their variable part but $3x$ and $4x^2$ are not like terms because one has an $x$ and the other has an $x^2$.

Some examples he looks at are:

1. $5x+3x\quad$ $5x$ and $3x$ are like terms.
2. $-8x^2y^3+10x^3y^2\quad$ $-8x^2y^3$ and $10x^3y^2$ are not like terms.
3. $9x^5y+5yx^5\quad$ $9x^5y$ and $5yx^5$ are like terms.

Next he talks about combining like terms. As you see in the examples above, there's a plus between the terms. You can actually combine them together using addition or subtraction IF they have like terms. To do this you add or subtract the coefficient, or number part of the term, and leave the variables alone. Looking at the examples above we get:

1. $5x+3x=8x$
2. Not like terms so cannot combine the two. You would have to leave it how it is.
3. $9x^5y+5yx^5=14x^5y$

He then does some examples with combining like terms.

1. $7x+4x=11x$
2. $5x^{23}-12x^{23}=-7x^{23}$
3. $12y+8x-2y-17x=10y-9x$
4. $\frac25x-\frac14+\frac3{10}x-\frac52=\frac7{10}x-\frac{11}4$

Combining like terms is when you combine all the terms with the same variable (and variable exponent) together. This includes terms with no variables!

Example

We want to combine like terms in this equation: $$3x+x-5+y-7y+4x+10=1$$

Notice that there are terms with $x$, terms with $y$, and terms with no variable at all. To combine all the like terms, we'll combine all the terms with $x$ into one, all the terms with $y$ into a different one, and finally all the terms with no variable (the constants).

Combining the $x$'s: $$\color{#8925AE}{3x+x}-5 + y - 7y\color{#8925AE}{+4x}+ 10=1\\ \color{#8925AE}{3x+x+4x} - 5 + 10 + y - 7y=1\\ \color{#8925AE}{8x} -5+10+y-7y=1$$

Combining the $y$'s: $$8x - 5 + 10 \color{#29A603}{+y- 7y}=1\\ 8x - 5 + 10 \color{#29A603}{- 6y}=1$$

And finally, combining the constants: $$8x \color{#1584D0}{ -5+ 10}- 6y=\color{#1584D0}{1}\\ 8x \color{#1584D0}{ -5+ 10-1}- 6y=0\\ 8x \color{#1584D0}{+4}-6y=0$$

And you're done! Our simplified equation is: $$\boxed{8x + 4 - 6y=0}$$

Here is an image with the previous example!