Problem 89 - Introductory Algebra for College Students

Question

Exercises \(89-91\) will help you prepare for the material covered in the next section. The sum of two numbers, \(x\) and \(y,\) is \(28 .\) The difference between the numbers is 6 a. Write a system of linear equations that models these conditions. b. Solve the system and find the numbers.

Step-by-Step Solution

Answer

The numbers are \(x=17\) and \(y=11\).

Step 1: Formulate the system of equations

Based on the description, two equations can be formulated: 1. \(x + y = 28\) - this describes the sum of the numbers. 2. \(x - y = 6\) - this describes the difference between the numbers.

Step 2: Solve the system of equations

These two equations are set to solve simultaneously. The easiest way is addition method. By adding the two equations: \(x + y = 28\) \(+\) \(x - y = 6\) We get: \(2x = 34\) which simplifies to \(x=17\). Substituting \(x=17\) into the equation \(x + y = 28\) we find \(y=28-17=11\).

Key Concepts

Solving Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that are solved together. They share the same solutions because they represent constraints on the same set of variables. In our exercise, the variables are represented by two unknown numbers, which we'll call \( x \) and \( y \). To solve these equations, we need to find values for \( x \) and \( y \) that satisfy both equations. This is known as solving the equations simultaneously.

• The goal is to find the values of all unknowns so both equations hold true.
• There are several methods to tackle these problems, such as substitution or the graphical method.
• In this problem, we have chosen the addition method because it's simple and straightforward.

The equations given are: - \( x + y = 28 \) - \( x - y = 6 \) The task is to find numbers \( x \) and \( y \) such that both conditions, the sum and the difference, are satisfied.

Addition Method

The addition method, also known as the elimination method, is a popular technique for solving simultaneous linear equations. It involves modifying the equations to eliminate one variable, making it easier to solve for the other. In our case, the method can be easily applied because the equations are already set up for it. Here’s how it works:

• We have two equations: \( x + y = 28 \) and \( x - y = 6 \).
• By literally adding the two equations, the \( y \) terms cancel out: \[ (x + y) + (x - y) = 28 + 6 \]
• This simplifies to \( 2x = 34 \). Now we just solve for \( x \): \[ x = \frac{34}{2} = 17 \]

Once \( x = 17 \) is found, we substitute it back into one of the original equations, say \( x + y = 28 \). This gives us: \[ 17 + y = 28 \]Solving for \( y \) yields \( y = 11 \). Thus, by using the addition method, we've found the solution \( x = 17 \) and \( y = 11 \). It’s an effective approach because it simplifies the equations significantly, requiring fewer steps than other methods.

Algebraic Models

Algebraic models are a way to express real-world situations using algebraic equations. They are immensely helpful in solving practical problems and making sense of complex relationships. In this exercise, the conditions were expressed using basic linear equations. The process of modeling involves translating words into equations. Here’s how our problem is modeled:

• The statement "The sum of two numbers is 28" translates to \( x + y = 28 \).
• The statement "The difference between the numbers is 6" becomes \( x - y = 6 \).

These algebraic models serve as a framework for problem-solving. By creating a system of linear equations, we can methodically find a solution, offering clarity to the problem at hand. In many real-life applications, algebraic models help in fields ranging from engineering to economics, where decision-making often requires understanding relations between different variables.Ultimately, these models simplify the complexity of real-world situations, allowing us to solve them symbolically and logically through algebra.