Problem 36 - Introductory Algebra for College Students

Question

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 2 x-5 y=-1 \\ 2 x-y=1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution of the system is \({(x, y) : x = 0.25 , y = -0.5}\).

Step 1: Equation Rearrangement

Rearrange the equations to align like terms. Here, both equations are already in the desired form

Step 2: Apply the Addition Method

To apply the addition method, add or subtract the equations to eliminate one of the variables. Here, the equations would be subtracted to eliminate variable y: \((2x - 5y) - (2x - y) = -1 - 1\). This simplifies to: \(4y = -2\).

Step 3: Solve for y

Solve for y by dividing both sides by 4: \(y = -2 / 4 = -0.5\).

Step 4: Substitute y Value into One of the Equations

We substitute y = -0.5 into the second equation to solve for x: \(2x - (-0.5) = 1\), which simplifies to: \(2x + 0.5 = 1\). Subtracting 0.5 from both sides gives: \(2x = 0.5\). Dividing by 2, we find that \(x = 0.5 / 2 = 0.25\).

Step 5: Write the Solution Set in Set Notation

Finally, we represent the solution in set notation: \({(x, y) : x = 0.25 , y = -0.5}\)

Key Concepts

Understanding Systems of Equations

A system of equations consists of two or more equations that share a common set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously. Systems of equations are often used to model real-world situations where multiple conditions must be met at once. Consider it like solving a puzzle where each equation offers some pieces of the solution.
When dealing with these systems, it's crucial to:

• Identify the number of equations and variables involved.
• Decide which method to use for finding the solution.
• Understand the relationship and interaction between the equations.

There are several methods to solve systems of equations, such as substitution, elimination (or addition method), and graphing. The choice depends on the specific system and context.

Finding the Solution Set

The solution set of a system of equations is the collection of all possible solutions that satisfy each equation within the system.
It is important to:

• Consider if the system has one solution, no solutions, or infinitely many solutions.
• Express the solution set using standard set notation, which provides a clear and structured way to present the solutions.

For instance, in our exercise, the solution set is expressed as \(\{(x, y) : x = 0.25 , y = -0.5\}\). This means the only set of values for \(x\) and \(y\) that satisfy both equations simultaneously is \(x = 0.25\) and \(y = -0.5\). Understanding the expression of solution sets helps in visually and logically comprehending the outcomes.

Exploring Linear Equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. They produce straight lines when graphed. In a system of linear equations, each equation has a linear form, typically represented as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Characteristics of linear equations include:

• They involve no products or powers of the variables.
• They graph as straight lines that either intersect, are parallel, or coincide with one another.
• They have at most one solution if represented by two distinct lines that intersect at a single point.

In our given exercise, both equations are linear, and the technique applied was the addition method. By aligning and combining these equations, one of the variables was eliminated, leading to a solution of the system. This method is particularly useful in making complex problem-solving simpler and efficient.