Video: One-Step Equations: Addition and Subtraction

In this video, we will learn how to write and solve one-step addition and subtraction equations in questions including word problems.

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Video Transcript

In this video, we will learn how to write and solve one-step addition and subtraction equations, paying close attention to the addition and subtraction property of equality. Before we start writing and solving equations, let’s think about what we know about an equation.

An equation is a mathematical statement that says something is equal to something else. For example, we could say that five is equal to five. If we were to take this positive three block and put it on the left side of this equation, all of a sudden the scale would be unbalanced. You would have eight on the left and only five on the right. But if we took a positive three block and put it on the right, the scale comes back into balance as both sides are equal to eight. This illustrates the addition property of equality. It tells us if you add the same value to both sides of an equation, the sides remain equal. This also applies to subtraction. Going back to our five equals five, if we subtract three from both sides of the equation, we end up with two on either side and the equation is balanced. And so, we can say If you subtract the same value from both sides of an equation, the sides remain equal.

Let’s consider one other property: the inverse property of addition. This tells us that any number added to its opposite will equal zero. The opposite of negative three is positive three. That means if we add three to both sides of the equation, on the left, we would have five plus zero, which in the end is just equal to five. And the same thing would happen on the right. Five plus zero equals five. All three of these properties are tools that we’ll need to help us solve one-step equations.

But what does that mean, “solve the equation”? If we have an equation like this, 𝑥 plus three equals five, we have an unknown value that’s being represented by a variable 𝑥. To solve the equation, we find the unknown value that makes the statement true. If we’re wondering what plus three equals five, we know that two plus three equals five. And so, we say that 𝑥 equals two. Here, we would call two the solution of the equation. The solution is the unknown value that makes the statement true. And that means a question might ask you to solve the equation. Or it might ask, what is the solution of the equation?

So let’s consider an example.

Given that five plus 𝑛 equals negative five, find the value of 𝑛.

We have the equation five plus 𝑛 equals negative five. This is telling us that when we add some value to five, the result is negative five. To find out what 𝑛 is, we need to think about the inverse property of addition. The inverse property of addition says that any number added to its opposite will equal zero. Since five and 𝑛 are being added together, if we subtract five, then five minus five will equal zero. But we also know the subtraction property of equality, which tells us that if we subtract five from one side of the equation, we must subtract five from the other side of the equation to keep it balanced. If five minus five is zero, then on the left side of the equal sign, we only have the variable 𝑛. And on the right side, we have negative five minus five, which equals negative 10. And so, we found that 𝑛 equals negative 10.

If you start with five and add negative 10 to that five, you’ll get negative five. This is a vertical method for solving the problem. We do this by writing the next line directly under the line above it. The other way to do it would be to use a horizontal method. We began with five plus 𝑛 equals negative five. But then, in the next line, we’ll add the negative five in the row with our original equation. So that it says five minus five plus 𝑛 equals negative five minus five. Both methods show that 𝑛 is equal to negative 10.

Here’s another example.

Solve for 𝑥: 𝑥 plus 4.8 equals negative 8.9.

This equation tells us that some value when we add 4.8 is equal to negative 8.9. When we see the word “solve” here, it means that we want to know which value for 𝑥 makes this statement true. If the equation has add 4.8, to solve it, we’ll need to do the inverse of adding 4.8. The inverse property says that any number added to its opposite will equal zero. But we also know, based on the subtraction property of equality, that if we subtract 4.8 from the left side of the equation, we must subtract 4.8 from the right side of the equation to make sure they stay equal. We now have on the left 𝑥 plus zero. We can just write that as 𝑥. And then, we have negative 8.9 minus 4.8, which equals negative 13.7. If we’ve calculated this correctly, we can plug in negative 13.7 in for 𝑥. And we’ll get a true statement.

We need to know is negative 13.7 plus 4.8 equal to negative 8.9. If we want to add 4.8 to negative 13.7, we subtract the two values first, which gives us 8.9 and then it takes the sign of the larger number. 13.7 was negative. So, the final answer will be negative. Negative 13.7 plus 4.8 does equal negative 8.9, which means the solution of 𝑥 is negative 13.7.

For the next example, we’ll have to first write an equation and then solve it.

William is playing a board game. From start, he moves 10 spaces forward. In his next turn, he moves six spaces back. How many spaces away from start is he now?

If William begins on start and he moves 10 spaces forward. On his next turn, he moves six spaces back. We want to know how many spaces is he away from start. For this unknown value, we can use the variable 𝑥. What kind of equation can we write to model the situation? We can write it a few different ways. First, William went forward 10 spaces. So, we can start with positive 10. And then, he went back six spaces. We can represent that mathematically with negative six. If you take positive 10 and subtract six, you’ll get 𝑥, the number of spaces away he is from the start. 10 minus six equals four. And that means our 𝑥-value is four.

Currently, the way it’s written: it says four equals 𝑥. But it’s fine to rearrange it in a more common way: 𝑥 equals four. 10 minus six equals 𝑥 is only one way to model this situation with an equation. We could say that 𝑥 — the number of spaces away from start William is — plus the six places backwards he walked must be equal to the 10 total forward spaces. If 𝑥 plus six equals 10, then we can solve the problem by subtracting six from both sides of the equation. 𝑥 plus six minus six equals 𝑥 plus zero, which is 𝑥, and 10 minus six equals four. Both methods show us that William is four places away from start.

Here’s another word problem example.

In 2016, Mississippi and Georgia had a total of 21 electoral votes. If Mississippi had six electoral votes, solve the equation six plus 𝑔 equals 21 to find the number of electoral votes that Georgia had 𝑔.

Here, we’ve actually been given an equation: six plus 𝑔 equals 21. And our goal is to solve for 𝑔. To solve for 𝑔, we want to get the 𝑔 by itself. We want to isolate this 𝑔 variable. To do that, we’ll need the inverse property of addition, which tells us that any value if you add its opposite equal zero. To get 𝑔 by itself, we want to get rid of the six. And we can do that by subtracting six. Six minus six equals zero. But we also know that to keep both sides of this equation equal, if we subtract six from one side, we must subtract six from the other side. On the left, it would be zero plus 𝑔. Or we can write that simply as 𝑔. And on the right, 21 minus six equals 15. And so, we found that 𝑔 equals 15. It’s worth checking to make sure that that’s true. Is six plus 15 equal to 21? Yes, it is. This tells us that 𝑔 is in fact 15. So, Georgia had 15 votes and Mississippi had six votes.

We’ll consider one final example.

A diver began his ascent to the surface. He ascended 20 meters to his next decompression stop and must ascend another 32 meters to return to the surface. Write and solve a subtraction equation to find the diver’s original depth before he started ascending.

The diver started at a depth that we don’t know. We can call this step 𝑥. We do know that the diver ascended 20 meters and still needs to ascend 32 meters before the diver reaches the surface. We want to write and solve a subtraction equation to find 𝑥, the diver’s original depth. If we began with 𝑥, the diver is 𝑥 meters deep and he rises 20 meters. Mathematically, we need to represent that with subtraction because he is not as deep as he was before. Even though he’s going up, he is becoming less deep. It’s his original depth minus the 20 meters he rose. And 𝑥 minus the 20-meter ascent will equal the 32 meters that remain, which means one equation is 𝑥 minus 20 equals 32.

To solve this equation, we add 20 to both sides. 32 plus 20 equals 52. And so, we can say that the diver’s original depth was 52 meters. We’ve written a subtraction problem. And we’ve solved for the missing depth to show that the diver’s original depth before he started ascending was 52 meters.

We’ll now think about the key points that are needed to solve one-step equations with addition and subtraction. Solving equations is finding the value or values of the variable that make the equation true. These values that make the equation true are called solutions. To solve equations, we use the addition and subtraction property of equality, which tells us if we add or subtract something from one side of the equation, we must do the same thing to the other side of the equation to make them remain true. And finally, we know the inverse property of addition, which tells us if we add the opposite value to a value, the result is zero. Using these key points, you can write and solve one-step equations with addition and subtraction.

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