Video: Using Linear Equations to Solve Problems

Fifty-two more than one-seventh of a number is thirty-nine less than the product of two and the number. What is the number?

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Video Transcript

52 more than one-seventh of a number is 39 less than the product of two and the number. What is the number?

Let’s begin by dissecting the sentence. So we begin with “52 more than.” This means we’re adding 52 to something. So we’ve 52 more than one-seventh of a number. Do we know what that number is? No, so we can call it 𝑥. And we want one-seventh of 𝑥. And the word “of” in mathematics means to multiply. So we want one-seventh times 𝑥. And again, we want 52 more than that.

Next, it says “52 more than one-seventh of a number is.” “Is” means we need an equal sign. “is 39 less than,” so if we want 39 less than something, we need to take 39 away from something. So we’re subtracting 39 from something. So it says “39 less than the product of two and the number.” So “product” means to multiply. So we want to multiply two and the number. And that number we’re talking about we let be 𝑥.

So let’s go ahead and read through the sentence one more time. 52 more than one-seventh of a number is 39 less than the product of two and the number. So now we need to solve for that number, meaning we need to solve for 𝑥. So let’s go ahead and get 𝑥 on both sides of the equation.

So let’s subtract one-seventh 𝑥 from both sides of the equation. The one-sevenths will cancel on the left, but we need to take two and subtract one-seventh. And in order to do that, we need a common denominator.

So if two really has a denominator of one, to make it be a seven, we have to multiply by seven. And whatever we do to the denominator, we have to do to the numerator. So we need to multiply two by seven and one by seven. So two turns into fourteen sevenths, and then we’re subtracting one-seventh.

So when we have a common denominator, we keep that denominator and we add or subtract the numerators. So here we’re subtracting, so 14 minus one, so we get thirteen sevenths. So on the right-hand side of the equation, we must have thirteen sevenths 𝑥. So we need to bring down the negative 39 and the 52.

Our next step to solve for 𝑥 is we need to add 39 to both sides of the equation. They cancel on the right. And then on the left, 52 plus 39 is 91. Next, we need to multiply both sides of the equation by seven. That way, seven is no longer on the denominator. And seven times 91 is 637.

So now our last step would be to divide both sides of the equation by 13. And 637 divided by 13 is 49. So 𝑥 is equal to 49. And 𝑥 represented the number we were trying to find. Therefore, 49 is our final answer.