Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. Find two numbers whose sum is 28 and difference is \(13 .\)

Step-by-Step Solution

Answer

The two numbers are 20.5 and 7.5.

Step 1: Define the Variables

Let's denote the two numbers we are looking for as \(x\) and \(y\). Here, \(x\) is one number, and \(y\) is the other number.

Step 2: Set Up the Equations

Using the problem statement, we can set up two equations. The first equation is based on their sum: \(x + y = 28\). The second equation is based on their difference: \(x - y = 13\).

Step 3: Solve the System of Equations

To solve the system, we can add the two equations together: \((x + y) + (x - y) = 28 + 13\). This simplifies to \(2x = 41\), so \(x = \frac{41}{2} = 20.5\).

Step 4: Substitute Back to Find the Second Variable

Now that we have \(x = 20.5\), substitute this back into the first equation: \(20.5 + y = 28\). Solving for \(y\) gives \(y = 28 - 20.5 = 7.5\).

Step 5: Verify the Solution

Check the solution by substituting \(x\) and \(y\) back into both original equations. For \(x + y = 28\), we have \(20.5 + 7.5 = 28\), which is true. For \(x - y = 13\), we have \(20.5 - 7.5 = 13\), which is also true.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed on a coordinate plane. They are written in the form: \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. These equations describe the relation between two variables, showing how they are connected. When working with linear equations, one can set up a problem to find unknown values, such as in systems of equations. This involves multiple equations that are solved together. In a system of linear equations with two variables, like those given in the original problem, the equations are usually connected to one another. By solving them, we can find the exact values of the variables that satisfy both equations simultaneously. Primary characteristics of linear equations include:

• They graph as straight lines
• Include no math operations other than addition, subtraction, and constant multiplication
• The degree of the variables is always 1

Solving Equations

Solving equations means finding the values of the variables that make the equation true. It is like uncovering the answer to a puzzle using the information given. There are various methods to solve equations, especially systems of linear equations. For example, in this exercise, we used two linear equations derived from the problem statement. The sum \( x + y = 28 \) and the difference \( x - y = 13 \). The goal is to find the values of \( x \) and \( y \) that satisfy both equations. One common approach is:

• **Substitution Method**: Solve one equation for a variable and substitute it into the other.
• **Elimination Method**: Add or subtract equations to eliminate one variable, making it easier to solve.

In the step-by-step solution, the elimination method was used. By adding both equations, variables\( y \) are eliminated, helping find \( x = 20.5 \). Then, by substituting \( x \) into one of the original equations, \( y = 7.5 \) is found.

Variables in Equations

Variables in equations act as placeholders for unknown values that we need to determine. They are often denoted by letters such as \( x \), \( y \), or \( z \). These variables are part of what makes solving equations both challenging and rewarding, as they represent the unknowns to be solved. In the context of a system of linear equations, these variables illustrate the relationship indicated by each equation.

• **Independent Variable**: A variable that stands alone and is not dependent on other variables.
• **Dependent Variable**: A variable that depends on the value of another variable in the equation.

In our exercise, \( x \) and \( y \) are variables whose values we found by setting up the equations. Solving a system with two variables requires identifying how each variable relates to the other, which can beautifully demonstrate the interplay between different parts of a mathematical system. Knowing how and when variables represent different things in an equation is key to understanding how to apply them to solve real-world problems.