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Video: Solving Two-Step Linear Equations over the Set of Natural Numbers

Bethani Gasparine

Find the solution set of 𝑥/4 + 8 = −2 in ℕ.


Video Transcript

Find the solution set of the equation 𝑥 divided by four plus eight equals negative two, in the natural numbers. Our possible answers are: A) the set of negative forty, B) an empty set called no solution, C) the set including the number twenty-four, and D) the set including the number negative ten.

Before we begin, let’s talk about what the natural numbers are. Natural numbers are your positive counting numbers. So numbers such as one, two, three, four, five. So when we solve our equation, 𝑥 divided by four plus eight equals negative two, our answer needs to belong in that set of natural numbers. Essentially, it needs to be positive.

In order to solve for 𝑥, we must first subtract eight from both sides of the equation. The eights on the left-hand side cancel, and on the right-hand side of the equation, negative two minus eight is negative ten. Now I must get rid of the four. And to do so, we must multiply both sides of the equation by four. This way, on the left-hand side, the fours cancel. And on the right-hand side, we take negative ten times four which is negative forty. So our final answer is: 𝑥 equals negative forty. But there is an issue. 𝑥 equals negative forty does not fall in the natural numbers.

Since our answer is negative, our solution set is empty. The number negative forty does not belong in the set of natural numbers. So again, the solution set is empty, written as the no solution. So our answer is B.

When we began this problem, we actually could’ve eliminated two of our options. Since we knew our solution set needed to be in the natural numbers, we couldn’t have a negative answer. So we could’ve crossed out A, negative forty, and D, negative ten, leaving us only two options to choose from. But using the same calculations in work, we would still find our answer to be B, no solution. It’s an empty set.