In Exercises 43–46, let x represent one number and let y represen...

Question

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is 24. Find the numbers.

Accepted Answer

Welcome back. I am so glad you're here. We are given some information and we're to create a system of equations based on that information. We're told that X should stand for one number and why for the other then? We need to solve for X and Y. After we've created our system of equations. Alright. The first bit of information we're given is that the result of addition of the first number? And the other number while the first number Is X. The other number is why we're adding them together is equals 14. There's our first equation in our system of equations. The second equation we're told that the product of two numbers that means we multiply them together, x times y is equals 48. Well there are two equations now we need to solve for X and Y. There are several ways to do this. I am going to use substitution. I'm going to subtract Y from both sides of the first equation and that will give me X equals 14 minus Y. And I can substitute this 14 minus Y wherever there's an X in the second equation. So now the second equation will read 14 minus Y instead of X times Y equals 48. Now to solve for why I'm going to distribute that Y to the 14 minus Y. So that will be 14. Why minus Y squared equals 48. And because this is a quadratic I want to get the Y positive. So I'm going to add it to both sides and I want to set one side equal to zero. So I'm also going to subtract that 14 Y from both sides bringing everything over to the right. But I'm actually going to write it on the left because it's just more traditionally done that way. So it's y squared minus 14. Y plus 48 equals zero. Now I can factor this, Y squared minus 14. Y plus 48 equals zero. What times what gives me? Why square? That's why times why? What two numbers when multiplied, Give me a positive 48 but add up to a negative 14. That would be a minus six minus eight. Now I can set both of these factors equal to zero to solve for y. So on the first one adding six table sides and on the second one adding eight to both sides and that will give me y equals six and y equals eight. Well I've got two solutions for y here. So yay for having solutions. But interestingly enough they've just got two answers for each of our different answer choices. So well let's plug these in and see what we get for our X values. Let's work with the first equation, X plus Y equals 14. And let's plug in our y equals six solution. So this will be X plus not y but six equals 14. Subtracting six from both sides, we get X equals 14 minus six is eight interesting. So when y equals six, X equals eight. Let's see what happens when we plug in the eight, that would be X plus eight equals 14. Subtracting the eight from both sides. And that gives us X equals 14 minus eight. Is six. There we go. So when Y equals eight, X equals six. So why is this true? Well, if we look at our two original Equations, we have addition and multiplication. And both of those are communicative, the order doesn't matter. So either the X or the Y could be eight or either the X. Or the Y could be six. If the X is six, the Y is going to be eight. If the X is eight, the Y is going to be six. And that's why our answer choices don't say X equals or Y equals. Because the order doesn't matter with the community of properties of addition and multiplication. So we look at our answer choices and this matches with answer choice. A Well done. We'll catch you on the next one.

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is 24. Find the numbers.

Key Concepts

System of Nonlinear Equations

Sum and Product of Numbers

Solving Nonlinear Systems

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