/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Let \(x\) represent one number a... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 43

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.

Short Answer

Expert verified
The numbers are 2 and 8.

Step by step solution

01

Formulate the System of Equations

From the conditions, the following equations can be formed: \n \(x + y = 10 \) (1) \n \(xy = 24\) (2)
02

Express One Variable in terms of the Other

From equation (1), we can express \(y\) in terms of \(x\): \n \(y = 10 - x\) (3)
03

Substitute Equation (3) into Equation (2)

We will now substitute equation (3) into equation (2) which leads to a quadratic equation: \n \(x(10 - x) = 24 \) \n \(-x^2 + 10x - 24 = 0\)
04

Solve the Quadratic Equation

The roots of the quadratic equation are the solutions for \(x\). Using the quadratic formula, we find: \n \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-10 \pm \sqrt{(10)^2 - 4*(-24)}}{-2}.\) \n This yields two solutions: \(x = 2\) and \(x = 8\).
05

Find Corresponding Values of Y

Substitute \(x\) values into equation (3): \n For \(x=2, y = 10 - 2=8.\) \n For \(x=8, y = 10 - 8=2.\)

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