Problem 5 - 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. $$\left\\{\begin{array}{r} x+2 y=7 \\ -x+3 y=18 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set to the given system of equations is \{-3, 5\}.

Step 1: Reformat the equations

Rearrange the equations to get them in similar formats. Here, by multiplying the second equation by -1, it can be made similar to the first equation. The equations would look like this:\[\left\{ \begin{array}{r} x+2y=7 \ x-3y=-18 \end{array} \right.\]

Step 2: Add the equations

By adding these equations directly, the value of y can be easily found:\[x+2y + x-3y = 7-18\]This simplifies to \(2x - y = -11\). Solving for y gives \(y = 2x + 11\)

Step 3: Substitute y back into an equation

Substitute y into one of the original equations and solve for x: Substituting into the first equation \(x + 2(2x+11) = 7\),Gives \(x + 4x + 22=7\),Which simplifies to \(5x = -15\),Thus, \(x = -3\)

Step 4: Find y

Substitute x = -3 into the equation found in step 2 for y:\[y = 2*(-3) + 11 = 5\]

Step 5: Write the answer in set notation

The solution for the system of equations is x = -3 and y = 5. In set notation, this is expressed as \[\{-3, 5\}\]

Key Concepts

Understanding a System of Equations

A system of equations is essentially a set of two or more equations that have common variables. Solving these means finding values for the variables that make all equations true at the same time. In our example, the system is composed of the equations:

• \(x + 2y = 7\)
• \(-x + 3y = 18\)

These equations share the variables \(x\) and \(y\). The goal is to find a pair \((x, y)\) that satisfies both equations simultaneously. In real-world scenarios, systems of equations can represent various dependent relationships, like finding equilibrium points in economics or determining intersection coordinates in geometry.
Using graphical, substitution, or addition methods are common ways to approach solving these systems. Here, we focus on the addition method.

Set Notation for Solutions

Set notation is a formal way to represent collections of objects or numbers. In the context of a system of equations, set notation is used to express the solution as a pair that satisfies all equations in the system. It is efficient for noting the results succinctly.
For instance, once you compute that \(x = -3\) and \(y = 5\) solve our example system of equations, you can represent the solution in set notation as

• \(\{-3, 5\}\)

This set neatly packages the solution into one easily understandable line. Set notation is especially useful when dealing with larger sets of solutions, such as when a system has infinite solutions.When no solution exists, the set is often denoted as an empty set \(\emptyset\) or "no solution" written out.

Solving Equations: The Addition Method

The addition method, also known as the elimination method, is a handy technique for solving systems of equations. In this method, you manipulate the equations to undo one of the variables. By doing this, you can solve for the remaining variable through addition.
Here’s the basic idea:

• First, adjust the equations so the coefficients of one of the variables are opposite. This often involves multiplying one or both equations.
• Next, add the equations together. This eliminates one variable from the system.
• Finally, solve the resulting equation for the remaining variable.

For our dataset, we adjusted the second equation by multiplying it by \(-1\) and added the resulting equations:\[(x + 2y) + (x - 3y) = 7 - 18\]This solved to simplify into one equation: \(2x - y = -11\). Further solving gave \(y = 5\) and \(x = -3\). This straightforward method helps reduce error and provides a clear path to the solution.