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 4 and 6.

Step 1: Formulate the Equations

The problem provides two pieces of information that can be written as equations. The sum of the two unknowns equaling 10 can be written as \(x + y = 10\). And, their product being 24 can be written as \(xy = 24\). Now, we have our system of nonlinear equations.

Step 2: Solve for one variable

Choose one of the equations (preferably the simplest one) and solve for one variable. From \(x + y = 10\), we can solve for \(y\) to get \(y = 10 - x\).

Step 3: Substitute to find the other variable

Now, substitute \(y\) from the equation \(y = 10 - x\) we got in Step 2 into \(xy = 24\) to solve for \(x\). That is \(x(10 - x) = 24\). Now simplify to get \(10x - x^2 = 24\). Rearranging to make it a quadratic equation, we get \(x^2 - 10x + 24 = 0\).

Step 4: Solve the quadratic equation

The equation \(x^2 - 10x + 24 = 0\) is a quadratic equation and can be solved by using quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) which when applied gives \(x = \frac{10 \pm \sqrt{10^2 - 4 * 1 * 24}}{2*1}\), computing this, we get two solutions for \(x\), namely, \(x = 4\) or \(x = 6\).

Step 5: Solve for the other variable

Substitute \(x\) into \(y = 10 - x\) to find corresponding values of \(y\). By doing this, we find \(y\) for the two possible values of \(x\). When \(x = 4\), \(y = 10 - 4 = 6\). When \(x = 6\), \(y = 10 - 6 = 4\). So, the pair of solutions are \((x = 4, y = 6)\) or \((x = 6, y = 4)\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree 2, typically taking the form \(ax^2 + bx + c = 0\). Quadratics often appear in problems involving areas, motions, and optimization tasks.
In this particular exercise, the equation \(x^2 - 10x + 24 = 0\) emerges from the given conditions. Understanding its roots is essential to solving systems where relationships form parabolas or contain squared terms.
Here’s how we solve this:

• Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -10\), and \(c = 24\).
• Substituting the values, calculate \(\sqrt{b^2 - 4ac}\) which simplifies the problem into finding the precise values for \(x\) that balance the equation.
• In many cases, especially when straightforward factorization isn't possible, the formula is a robust tool to find the roots of any quadratic.

This formula helps to find the necessary solutions that fit the problem's conditions, like providing the two numbers in this exercise.

Nonlinear Equations

Nonlinear equations are equations in which the unknown or its various expressions appear at powers other than one. Unlike linear equations, they form curves instead of straight lines when graphed.
In this exercise, we encounter the system of nonlinear equations where the conditions give rise to:

• The sum equation: \(x + y = 10\), which together with the
• Product equation: \(xy = 24\), combines linear and quadratic forms. The existence of these terms creates a nonlinear system of equations.

Such nonlinear setups require specific approaches, such as substitution or elimination methods, to unravel the interactions among their variables. The key is identifying how the involved terms contribute to the equation and applying appropriate methods to find the solution.

Substitution Method

The substitution method is a technique to solve systems of equations by expressing one variable in terms of the other and substituting it back into the remaining equations.
In this problem, the substitution method begins with the simpler equation, \(x + y = 10\), to express \(y\) as \(y = 10 - x\). By substituting \(y\) into the product equation, \(xy = 24\), we form a solvable quadratic equation:

• Substitute \(y\) in the function: \(x(10 - x) = 24\).
• Simplify to form a quadratic equation — \(10x - x^2 = 24\) — leading to the standard form \(x^2 - 10x + 24 = 0\).

The power of substitution lies in transforming more complex problems into a form where known solutions can be easily applied. It simplifies systems with more than one relationship by compactly solving one variable in terms of the other.