Problem 43 - 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 10 and their product is \(24 .\) Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 4 and 6.

Step 1: Formulate the System of Equations

Starting with the given conditions, these can be expressed as two equations. The sum of two numbers \(x\) and \(y\) is 10 which can be written as \(x + y = 10\). Their product is 24 which can be written as \(xy = 24\). So, the system of equations is: \[\begin{cases} x + y = 10 \ xy = 24 \end{cases}\]

Step 2: Solve the System of Equations

For a system of equations, an equation can be solved for one variable and then substituted into the other equation. Solving \(x + y = 10\) for \(x\), gives \(x = 10 - y\). Substitute this into the other equation, \(xy = 24\), gives: \((10 - y)y = 24\). Simplifying, will give a quadratic equation \(y^2 - 10y + 24 = 0\).

Step 3: Solve the Quadratic Equation

The quadratic equation can be solved either by factoring, completing the square or using the quadratic formula. The equation \(y^2 - 10y + 24 = 0\) can be factored to \((y - 4)(y - 6) = 0\). Setting each factor equal to zero gives the solutions \(y = 4\) or \(y = 6\).

Step 4: Find the Corresponding x Values

Having found the solutions for \(y\), the corresponding \(x\) values can be found by substituting \(y\) back into \(x = 10 - y\). For \(y = 4\), \(x = 10 - 4 = 6\). For \(y = 6\), \(x = 10 - 6 = 4\).