Problem 43 - College Algebra

Question

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.

Step-by-Step Solution

Answer

The numbers are 2 and 8.

Step 1: Formulate the System of Equations

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

Step 2: Express One Variable in terms of the Other

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

Step 3: 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 \(-x^2 + 10x - 24 = 0\)

Step 4: 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 \(x = 8\).

Step 5: Find Corresponding Values of Y

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