Problem 44 - Algebra and Trigonometry

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 20 and their product is \(96 .\) Find the numbers.

Step-by-Step Solution

Answer

The solutions to the system of equations are two pairs (14, 6) and (6, 14).

Step 1: Formulate the Equations

Transform the given conditions into mathematical equations. The conditions can be written as \(x + y = 20\) and \(xy = 96\).

Step 2: Use Substitution or Elimination to Solve

First, solve the first equation for either \(x\) or \(y\). In this case, it is easier to solve for \(x\). So, \(x = 20 - y\). Substitute this into the second equation we get \((20 - y)y = 96\). This turns the system of equations into a quadratic equation.

Step 3: Solve the Quadratic Equation

By simplifying the quadratic equation \((20-y)y = 96\) we get \(y^2 - 20y + 96 = 0\) . Now we can use the formula for solving quadratic equations, which is \(y = (b± √(b^2 - 4ac)) / (2a)\) . Here, \(a = 1, b = -20, c = 96\). By substituting \(a,b,c\) values, we get \(y = 20± √((-20)^2 - 4*1*96) / (2*1) = 14, 6\) .

Step 4: Solve for the Other Variable

To find the \(x\) values, substitute \(y\) values into the equation \(x = 20 - y\). When \(y = 14\) , \(x = 20 - 14 = 6\) and when \(y = 6\) , \(x = 20 - 6 = 14\) .