Problem 43 - 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. $$\left\\{\begin{array}{l} \frac{4}{5} x-y=-1 \\ \frac{2}{5} x+y=1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set is \( \{(0,1)\} \)

Step 1: Express the equations in more convenient form

The given system of equations can be rewritten by multiplying each equation by a suitable number so their coefficients are no longer fractions. This gives: \(\begin{array}{l} 4x-5y=-5 \ 2x+5y=5 \end{array}\)

Step 2: Apply the Addition Method

The Addition Method consists of adding or subtracting the equations to get a new equation in one variable. In this case when we add the two equations the y variable is eliminated: (4x-5y) + (2x+5y) = -5 + 5. This simplifies to 6x = 0.

Step 3: Solve for x

From the previous step, we obtained 6x = 0. We can then solve that for x by dividing both sides by 6 giving x=0.

Step 4: Substitute x into one of the original equations to solve for y

Setting \(x=0\) in \(\frac{2}{5}x + y = 1,\) and simplifying gives \(y=1\) .

Step 5: Express the solution in set notation

The solution set of the system of equations is described by the set of all x and y values that satisfy both equations. In this case, x=0 and y=1. This gives us the solution set: \( \{(0,1)\} \).

Key Concepts

Understanding a System of Equations

A system of equations consists of two or more equations with the same variables. When we talk about solving a system of equations, we mean finding values of the variables that satisfy all the equations in the system at the same time. In this exercise, our system is composed of two linear equations with variables x and y: \(\frac{4}{5}x - y = -1\) and \(\frac{2}{5}x + y = 1\). These equations intersect at a point, which represents the common solution to both equations. This intersection is a fundamental concept because it shows how different equations can work together to find a shared solution. In other words, the solution is like finding a point on a map where two paths cross. This approach is commonly used in mathematics to solve problems involving multiple equations.

Using Set Notation to Express Solutions

Set notation is a way to represent collections of objects or numbers that share a common property. When we find a solution to a system of equations, we can use set notation to express the solution neatly. In this exercise, the solution found was \(x = 0\) and \(y = 1\). Using set notation, we express this solution as \(\{(0, 1)\}\). This notation simply means that the pair \((x, y)\) equals \((0, 1)\), and is the exact pair that satisfies both equations. Set notation helps provide clarity and precision, making it easy to communicate the specific solution found to the system of equations. Additionally, it ensures that all potential solutions are clearly defined and easily understood by anyone reading the solution.

Solving Linear Equations with the Addition Method

The addition method, also known as the elimination method, is a way to solve a system of linear equations. The goal is to eliminate one of the variables by adding or subtracting the equations. In the given problem, the original equations were transformed to \(4x - 5y = -5\) and \(2x + 5y = 5\). By adding these modified equations, the y terms cancel out: - \((4x - 5y) + (2x + 5y) = -5 + 5\)- This simplifies to \(6x = 0\)Then, we solve for \(x\) by dividing both sides by 6, giving us \(x = 0\). Once \(x\) is determined, it is substituted back into one of the original equations to find \(y\). By substituting \(x = 0\) into \(\frac{2}{5}x + y = 1\), and solving, we find \(y = 1\). Using the addition method simplifies the process and makes it systematic to find the solution to linear equations. This method is particularly useful for systems with two equations because it quickly resolves one variable and simplifies the solving process for the other.