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Amy has a master's degree in secondary education and has taught math at a public charter high school.
What do we do normally when we have parentheses? We usually evaluate the inside of the parentheses first as we follow our order of operations. For example, if we have something like (3 + 1)5 + 2, we should first do the operation inside the parentheses and then evaluate the outside. So, we would do 3 + 1 to get 4 first before multiplying that result by the 5 to get (4)5 = 20. Now we can finish our problem by adding the 2 to get a final answer of 22.
But what if we add a variable into the mix so that we need to solve (3x + 1)5 + 2 = 0? What do we do then? We can't add the 3x + 1 together. The only way we could combine the 3 and the 1 is if the 3x and the 1 were like terms, which means that they share the same variable with the same exponents. As you can see, the 3 has an x for a variable, but the 1 doesn't. So, what do we do?
In algebra, we have a property to help us remove those parentheses. It's called the distributive property, and it tells us that we can remove a pair of parentheses by multiplying the term outside the parentheses with every term inside the parentheses. In the language of algebra, the distributive property looks like this: a(b + c) = a(b) + a(c), where a, b, and c are terms, either just numbers or numbers with variables, and the parentheses mean multiplication.
You can see that we've multiplied the term outside with every term inside the parentheses and we've kept the operation inside the parentheses between our multiplications. I like to think of the distributive property as distributing a hug to everybody inside the parentheses. If you think of parentheses as your arms, you can kind of see how it looks like a big hug.
If you have several terms inside the parentheses, it's like giving a group hug to everybody inside. If your arms weren't big enough to give a group hug to everybody, what can you do? You can go around and hug each individual term. Let's see this in action.
We have our equation (3x + 1)5 + 2 that we want to solve. We see parentheses with a couple terms inside. I think group hug, but my arms are too short. What do I do? I distribute my hug to each term to get (3x)5 + (1)5 + 2. Now I can go ahead and multiply each individual hug to get 15x + 5 + 2.
The next step in solving an equation like this is to collect like terms. I take my highlighter and I start by highlighting the 15x. I keep going to see if I have any more terms that also have an x. I don't see any, so now I choose a different color highlighter. I highlight the next term, the 5. I keep going to see if there are other numbers without variables. I see a 2, so I go ahead and highlight that with the new color. Now I've highlighted all my terms. The 0 I can ignore since the 0 doesn't change anything.
Now I go through and add my like terms. The 15x is by itself so there is nothing to add it and it stays the same. The next color highlight has the 5 and a 2, so I can add those together to get 5 + 2 = 7. Now I can rewrite my equation as 15x + 7 = 0. I've collected my like terms and I can move on to the next step of solving.
To solve, I need to isolate my variable to get it by itself on one side of the equation. Since the variable is on the left side, I'm going to move everything else to the opposite side. To do this, I first move things that are being added or subtracted. To move things over, I do the opposite of the operation. So if I have addition, I subtract. If I have subtraction, I add. I see that I have an addition of 7, so I will subtract 7 from both sides. Doing this I get 15x + 7 - 7 = 0 - 7, which turns into 15x = -7.
Now I can move things that are being multiplied and divided. Again, I do the opposite operation. If I see multiplication, I divide. If I see division, I multiply. I see multiplication of 15, so I will divide by 15 on both sides, 15x / 15 = -7 / 15. Doing this I get x = -7 / 15. This is also my answer since I can't simplify the -7/15 any further. If I had -5/15, then I could simplify it further to get -1/3, and this simplified form would be my answer. But -7/15 can't be simplified any further, so I know that I am done and this is my final answer.
Now, what have we learned? We've learned that to solve an equation involving parentheses, we can use the distributive property to help us remove our parentheses. This property tells us we can remove a pair of parentheses by multiplying the term outside the parentheses with every term inside the parentheses.
In algebra language, the distributive property will read as a(b + c) = a(b) + a(c). We can think of the distributive property as distributing a big group hug to every individual term inside the parentheses. We also remember that like terms are the terms that share the same variable with the same exponents, where terms are either numbers or numbers with variables.
The steps to solve an equation after we remove the parentheses require us to isolate our variable. To do this we start by moving those numbers that are being added or subtracted. I performed the opposite operation to move those numbers over. Next I work on moving those numbers that are being multiplied or divided. Again, I perform the opposite operation to move those numbers over. For every step, I remember that what I do to one side, I have to do to the other. Once my variable is isolated and all my numbers have been simplified, then I am done.
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