Video: Writing Equations and Solving Them Mentally

Learn how to write equations containing a variable (such as 𝑝 − 5 = 20) and solve them using mental math strategies (the number 5 greater than 20 is 25, so 𝑝 = 25). First, we explain what an equation is, then we run through some examples.

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Video Transcript

Today, let’s look at writing equations and then solving them mentally. But before we start trying to write equations, we probably should ask the question, what is an equation? An equation is a sentence that contains two expressions separated by an equal sign. Here’s an example of that. Seven equals eight minus one. Here’s another example. 18 equals 13 plus 𝑥. This equation contains a variable which is the 𝑥. This equation, 18 equals 13 plus 𝑥, is neither true or false until the variable is replaced with a number.

Let’s say we replaced 𝑥 with a three. The equation is now false because 18 is not equal to 16. But when we’ve replaced the 𝑥 with the five, it becomes a true statement. We say that the equation here is true when 𝑥 equals five. We also say that five is the solution of the equation. A solution of an equation is the numerical value for the variable that makes the sentence true. In this case, five makes our sentence true. Solving an equation is the process of finding a solution. When we try to figure out what the variable is equal to, we call that solving the equation. Now, let’s take a look at how we solve an equation with mental math.

So here’s an equation. 𝑝 minus five equals 20.

Remember that we wanna solve these problems mentally without writing things down. So we think some number take away five is equal to 20. And that number must be five more than 20. Therefore, 𝑝 must equal 25. 𝑝 equals 25 is the correct answer.

Here’s another example. If 𝑘 equals 60, is 28 plus 26 equal 𝑘 true or false?

And again we wanna use a mental math strategy to solve this question. So you think about 28 and 26. They have an eight and a six in their ones position. And 60 has a zero in its one position. Eight plus six equals 14. And that means the solution to 𝑘 is going to have to have a four in the ones position. At this point, we don’t know what 𝑘 is. But we know that if 𝑘 equals 60, then this sentence must be false. We can say for sure that 28 plus 26 does not equal 60, which makes this statement false. Now, we are going to look at a question that asks us to write an equation.

The total cost of a skateboard and knee pads is 247 dollars. If the skateboard costs 235 dollars, write an equation to find the cost of the kneepads, 𝑘. Then solve the equation.

The problem is asking us to do two things, write an equation and then solve the equation that we write. We’re given three different pieces of information. The total cost is 247 dollars. The skateboard costs 235 dollars. And the missing cost, the cost that we don’t know, is 𝑘, and that’s the cost of the kneepads. We know that the total cost equals the skateboard price plus the kneepads price. Let’s substitute these two pieces with the information that we already know. Total cost is 247. Skateboard costs 235. What should we put here for the knee pads price? Because we don’t know the knee pads price, we need to substitute a variable here. In our case, that’s where the 𝑘 goes. Our equation becomes 247 equals 235 plus 𝑘.

Step one of the problem is complete. This is an equation we can use to solve step two. And this part can be done mentally. What added to 235 would equal 247? Well, if you add ten, you get from 235 to 245 add two more to that and now you’re at 247. Put both of those numbers you added together, 10 and two, and you realise that 𝑘 must be equal to 12. 𝑘 is equal to twelve. And that means the kneepads cost 12 dollars. Our final answer here is 𝑘 equals 12 dollars.

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