Problem 66 - Intermediate Algebra

Question

Use a nonlinear system of equations to solve each problem. Number Problem. The sum of the squares of two numbers is \(221,\) and the sum of the numbers is 9. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 14 and -5.

Step 1: Define the Variables

Let the two numbers be denoted by \( x \) and \( y \). These are the variables we need to solve for.

Step 2: Setup the Equations

Based on the problem, we have two key pieces of information:1. The sum of the squares of the numbers is 221, leading to the equation: \( x^2 + y^2 = 221 \)2. The sum of the numbers is 9, leading to the equation: \( x + y = 9 \)

Step 3: Substitute to Eliminate y

Use the equation \( x + y = 9 \) to express \( y \) in terms of \( x \):\[ y = 9 - x \]Substitute \( y = 9 - x \) into the first equation:\[ x^2 + (9 - x)^2 = 221 \]

Step 4: Simplify and Solve for x

Expand the equation:\[ x^2 + (9 - x)^2 = x^2 + (81 - 18x + x^2) = 221 \]Combine like terms:\[ 2x^2 - 18x + 81 = 221 \]Subtract 221 from both sides:\[ 2x^2 - 18x + 81 - 221 = 0 \]This simplifies to:\[ 2x^2 - 18x - 140 = 0 \]

Step 5: Solve the Quadratic Equation

Divide the whole equation by 2 to simplify:\[ x^2 - 9x - 70 = 0 \]Now, factor the quadratic equation:\[ (x - 14)(x + 5) = 0 \]Set each factor to zero and solve for \( x \): 1. \[ x - 14 = 0 \Rightarrow x = 14 \]2. \[ x + 5 = 0 \Rightarrow x = -5 \]

Step 6: Solve for y using x values

Substitute the values of \( x \) back into \( y = 9 - x \):1. When \( x = 14 \), \( y = 9 - 14 = -5 \)2. When \( x = -5 \), \( y = 9 - (-5) = 14 \)

Step 7: Verify the Solutions

Verify that the solutions satisfy both original equations:- For \((x, y) = (14, -5)\):\( 14^2 + (-5)^2 = 196 + 25 = 221 \) and \( 14 + (-5) = 9 \)- For \((x, y) = (-5, 14)\):\( (-5)^2 + 14^2 = 25 + 196 = 221 \) and \( -5 + 14 = 9 \)Both solutions are valid.

Key Concepts

Quadratic Equations

Quadratic equations are a vital part of algebra and they often come into play when working with nonlinear systems. In this exercise, we dealt with equations where the variables were squared, typical of quadratic equations. The general form is:

• \( ax^2 + bx + c = 0 \)

Here, we used an equation derived from the sum of squares of two numbers, \( x^2 + y^2 = 221 \). After substituting \( y = 9 - x \) from the sum equation \( x + y = 9 \), we obtained a standard quadratic form with respect to \( x \). Solutions to quadratic equations like these can be obtained via factoring, using the quadratic formula, or completing the square. In our solution, we factored the equation once it was simplified.

Algebraic Substitution

Algebraic substitution is a method used to simplify and solve systems of equations. It's especially useful where you have more equations than unknowns. In this problem, we used algebraic substitution to replace one variable with an expression involving the other variable. Starting from the equation:

• \( x + y = 9 \)

We expressed \( y \) as:\( y = 9 - x \). This was then substituted into the first equation \( x^2 + y^2 = 221 \), giving us a single equation in terms of \( x \) alone. This process of substitution reduces a system of equations to a single equation, thereby simplifying the problem.

Problem-Solving Strategies

Effective problem-solving strategies are centered around breaking down complex problems into simpler parts. In this exercise, dividing the problem into a series of strategic steps allowed for an organized approach:

• Define the variables to understand what you’re solving for.
• Set up the equations based on given conditions.
• Use substitution to simplify and reduce the equations.
• Factor or use other methods to solve the simplified equation.
• Back-substitute to find the corresponding variable.

By methodically following these steps, complex systems become much more manageable to solve. Each step builds upon the previous one, ensuring that no information is lost or overlooked.

Mathematical Verification

Verification is a crucial step in problem-solving. It reassures you that the solutions satisfy the original conditions set by the problem. In our exercise, once the solutions for \( x \) and \( y \) were found, they were verified against the initial equations. For example:

• For \((x, y) = (14, -5)\): \( 14^2 + (-5)^2 = 221 \) and \( 14 + (-5) = 9 \)
• For \((x, y) = (-5, 14)\): \( (-5)^2 + 14^2 = 221 \) and \( -5 + 14 = 9 \)

Both sets of results meet the criteria given in the problem. Verification provides confidence that the solutions are correct and applicable. It's a vital step to ensure any errors made in computation are caught and corrected.