Lesson Video: Two-Step Equations Mathematics • Sixth Year of Primary School

Video Transcript

In this video, we will learn how to solve two-step equations. To do that, we will use the balance method. That’s a general method for solving equations based on the additive and multiplicative properties of equality.

Before we try to solve this equation, let’s think about what it’s doing. This equation is giving us information about an unknown value. Here, that unknown value is represented with the letter 𝑥. The simple equation tells us that when this 𝑥-value has undergone some operations, it led to the result of 19. To solve this equation. We want to reverse all the operations that this 𝑥 has gone through. And we want to do it in the contrary order. That is, the opposite order that it happened in.

Our first term two 𝑥 tells us that some value is doubled. And whatever that amount was, you add five to it. And the result was 19. Two 𝑥 plus five equals 19. We can visualize this with a bar diagram. Two 𝑥 plus five is equal to 19. The bar diagram suggests that we could divide 19 into two pieces. If we took a piece that was five, the remaining portion on the left would be equal to 19 minus five, which is 14 because 14 plus five equals 19.

If we take away the part of the bar that represents five from the top bar and the bottom bar, the two bars are still equal. We have a bar that’s two 𝑥 in the top and a bar that equals 14 in the bottom. What if we divide both of these bars in half. Two 𝑥 divided by two equals 𝑥, and 14 divided by two equals seven. If two 𝑥 equals 14, then one 𝑥 equals seven. Going back to this statement, some value is doubled, then we add five, and it equals 19. When we double seven, we multiply it by two. Two times seven is 14, plus five equals 19.

To help us think about how to solve this mathematically, let’s visit the idea of a balance. On the left, we have two 𝑥 plus five. And on the right, we have 19. But inside that 19, there’s also a five. We know that 14 plus five equals 19. On the right then, we can switch out the 19 for one block of 14 and one block of five. If we take away five from both sides, we are subtracting five from both sides of our equation. And what will be left? On the left side, we’ll have two 𝑥. And on the right side, we’ll have 14.

If we divide two 𝑥 by two, we get 𝑥. And if we divide 14 by two, we get seven. Which tells us that 𝑥 equals seven. We have subtracted by five and divided by two, which is working in reverse order of the equation we were given. The equation we were given is multiplied by two and then add five. But why is this possible? Why does the balance method work? It seems really intuitive. But there are two main properties that this is showing. They’re properties of equality for multiplication and addition.

In the multiplication property of equality, we know that if both sides of the equation are multiplied by the same number or if both sides of the equation are divided by the same number, then the equality is still true. We might write it in symbols like this. If 𝑎 equals 𝑏, then 𝑎 times 𝑐 is equal to 𝑏 times 𝑐. Similarly, if the same amount is added to or subtracted from both sides of the equation, then the equality is still true. And written in symbols, if 𝑎 equals 𝑏, then 𝑎 plus 𝑐 equals 𝑏 plus 𝑐.

Let’s put these properties into action by looking at some examples.

A number is tripled and seven is subtracted from the result. If the answer is 17, what is the number?

The first thing we need to do here is take this statement and turn it into a mathematical one. We have an unknown number. And to represent an unknown number, we need a variable. It’s common to use the variable 𝑥. And our number is tripled. If we triple 𝑥, it becomes three 𝑥. Whatever this three 𝑥 is, seven is subtracted from that. Because we don’t know what three 𝑥 is, we have to write three 𝑥 minus seven. And the answer is 17. So, we can say that three 𝑥 minus seven equals 17.

If we start with this equation, three 𝑥 minus seven equals 17, we need to work in the reverse order. The last thing we did to our number was subtract seven. In order to reverse this operation, we need to add seven. But in order to keep both sides of the equation equal, we need to add seven to both sides based on the addition property of equality. On the left, three 𝑥 minus seven plus seven equals three 𝑥. And on the right, 17 plus seven equals 24.

To undo the subtract seven, we added seven to both sides. How would we then undo this tripled value? If we remember that triple means times three, to undo that, we would divide by three. We need to then divide by three on both sides. Three 𝑥 divided by three equals 𝑥. 24 divided by three equals eight. We can do this because of the multiplication property of equality. Which says if we multiply or divide by the same thing on both sides of the equation, the statement remains equivalent. Using these two properties, we’ve shown that 𝑥 equals eight.

In problems like this, it can be worth it to check and make sure that’s true. If our number is eight, and it’s tripled, eight times three equals 24. Seven is subtracted from that result. 24 minus seven should equal 17. And it does. So, we’ve correctly identified the missing number, which is eight.

Here’s another example.

Find the solution set of seven 𝑥 plus 11 equals negative 24 in the set of all integers.

We remember that this symbol that looks a little bit like a Z is the set of all integers. We know that seven 𝑥 plus 11 equals negative 24. And the solution set will be the values for 𝑥 that makes seven times 𝑥 plus 11 equal to negative 24. In order to solve for 𝑥, we need to get 𝑥 by itself. And the first thing we can do is subtract 11 from both sides of the equation. If we subtract 11 from both sides, it keeps this equation balanced.

Seven 𝑥 plus 11 minus 11 equals seven 𝑥. And negative 24 minus 11 equals negative 35. And now, we have seven 𝑥 equals negative 35. It means seven times some number equals negative 35. In order to undo that multiplied by seven, we need to divide both sides of this equation by seven. Seven 𝑥 divided by seven equals 𝑥. And negative 35 divided by seven equals negative five. This means that when we multiply seven by negative five and then add 11, we should get negative 24.

Seven times negative five is negative 35. Is negative 35 plus 11 equal to negative 24? It is. And so, we found the value that 𝑥 must be is negative five. We should be careful here because our question is asking us for a solution set. And that means we’ll need to use the curly brackets. But the only integer, the only value, for 𝑥 that makes this statement true is negative five. And that means that’s the only value that goes into the solution set. The solution set of this equation is negative five.

Here’s another example.

Find the solution set of the equation negative four 𝑥 plus three equals four.

We’re given the equation negative four 𝑥 plus three equals four. This negative four 𝑥 means negative four times 𝑥. And whatever that value is, we’re adding three to it. We find the 𝑥-value by isolating the 𝑥, by getting it by itself on one side of the equation. In order to get 𝑥 by itself, we first subtract three from both sides. Negative four 𝑥 plus three minus three equals negative four 𝑥. And four minus three equals one.

Because we now have negative four times 𝑥 equals one, to find 𝑥 by itself, we need to divide both sides of the equation by negative four, which you might write like this or in the fraction form like this. Negative four 𝑥 divided by negative four equals 𝑥. And we need to be careful on the right side because we have one divided by negative four. We can write that as a fraction, negative one-fourth, or in the decimal format, negative 0.25.

Once we found negative one-fourth for 𝑥, it’s worth plugging it back in to check. Negative four times negative one-fourth equals one. And one plus three does equal four, which means we found a correct value for 𝑥. But we should be careful here because the question is asking for a solution set. And that means we’ll use the curly brackets. There’s only one value in the solution set. And that’s negative one-fourth, which in set notation looks like this.

In the next example, we have 𝑥 being divided by something instead of multiplied by something.

Find the solution set of 𝑥 over four minus three equals negative five in the set of all integers.

First, we remember that this symbol is the set of all integers. And then, we can turn our attention to solving for 𝑥. If 𝑥 over four minus three equals negative five, we want to try to isolate 𝑥, to get 𝑥 by itself. And the first thing we should do in order to do that is add three to both sides. Once we do that, we’ll get 𝑥 over four is equal to negative two because negative five plus three equals negative two.

We know that 𝑥 over four means 𝑥 divided by four. Some value divided by four equals negative two. To find out what that value is, we need to do the opposite. If 𝑥 is being divided by four, we need to multiply both sides of the equation by four, like this. Four times 𝑥 over four equals 𝑥, and negative two times four equals negative eight. At this point, we should check and see if that’s true.

Is negative eight divided by four minus three equal to negative five? Negative eight divided by four is negative two. And negative two minus three is negative five. This means that 𝑥 is equal to negative eight. But here our solution needs to be given as a set notation. That’s the curly brackets. And the solution set for this equation is just negative eight.

In all of our previous examples, the first thing that we’ve done is add or subtracted a value. It’s not always the case that we do addition or subtraction first. Let’s look at this example and see why that might not be the case.

Solve 𝑥 plus three over five equals 10.

We know that 𝑥 plus three over five equals 10. And to solve for 𝑥, we need to get it by itself. We need to isolate it. And that means we need to undo the operations that are being done to 𝑥. In the numerator, we have some number 𝑥 plus three. And whatever that number plus three is is being divided by five. And that means to find out what that number is, we first need to undo this division.

And we can do that by multiplying by five. But if we multiply by five on the left, we need to multiply by five on the right. On the left, the five in the numerator and the five in the denominator cancel out, leaving us with 𝑥 plus three. And on the right, we have 10 times five, which is 50. And now, we have a simple equation, 𝑥 plus three equals 50. So, we subtract three from both sides. And we get 𝑥 equals 47.

Why couldn’t we subtract three from both sides? What would happen if we did? On the left, we would have 𝑥 plus three over five minus three. And on the right, we would have seven. Now, this is still a true statement. 𝑥 plus three over five minus three does equal seven. However, we’re no closer to solving for 𝑥. And that’s because the entire numerator is being divided by five. And we can’t take this plus three out of the numerator. This is because we have to reverse the operations in the contrary order than when we started.

Let’s do a quick recap on the key points for solving two-step equations. The balance method is a general method for solving equations based on the additive and multiplicative properties of equality. These properties state the following. If the same amount is added to or subtracted from both sides of the equation, then the equality is still true. And if both sides of the equation are multiplied by the same number or if both sides are divided by the same number, then the equality is still true. Based on this, we say that a simple equation can be solved by reversing all the operations in the contrary order to yield the missing value.