Problem 32 - Introductory Algebra for College Students

Question

Solve each system by the substinuion method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}3 x-4 y=x-y+4 \\\2 x+6 y=5 y-4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of equations is {(3, -2)}.

Step 1: Simplify The Equations

The given system of equations is: \(3x - 4y = x - y + 4\) and \(2x + 6y = 5y - 4\). Simplify these equations by grouping similar terms, resulting in: \(2x + 3y = 4\) and \(2x + y = -4\)

Step 2: Express One Variable in Terms of the Other

Solve the second simplified equation for \(x\). It becomes \(x = -4 - y / 2\)

Step 3: Substitute into the Other Equation

Substitute \(x\) from step 2 into the first simplified equation. The equation then becomes \(2(-4 - y/2) + 3y = 4\).

Step 4: Solve for the Variable

Solve the equation in step 3 for \(y\). After doing this, one solution for \(y\) is -2.

Step 5: Substitute the Solution for y into the Equation for x

Substitute \(y = -2\) into the equation \(x = -4 - y / 2\), to find the solution for \(x\) which equals 3.

Step 6: Represent the Solution in Set Notation

In set notation, the solution is represented as {(3, -2)}.

Key Concepts

System of Equations

When we talk about a system of equations, we're referring to a set of two or more equations that contain the same variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. This comes in handy for various real-world problems, where multiple factors need to be considered at the same time.

In mathematics, there are different methods to solve a system of equations, such as graphing, substitution, and elimination. The substitution method, which we used in the exercise, involves expressing one variable in terms of another, then substituting this expression into another equation. Thus, you solve a system step-by-step by reducing it to one equation with one unknown.

Here is a brief overview of what happens when solving systems of equations using the substitution method:

• Simplify the equations wherever possible.
• Solve for one variable in terms of the other variables.
• Substitute this expression into another equation in the system.
• Solve the new equation for the remaining variable.
• Finally, back-substitute to find the value of the first variable.

This process helps in breaking complex problems into simpler parts, making it easier to find the solution. In our example, we simplified the equations before making any substitutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). These expressions are fundamental in representing real-world situations and relationships between different quantities.

When working with systems of equations, we often deal with linear algebraic expressions, which can be simplified by combining like terms or rearranging terms to isolate variables. Consider the equation from the exercise:
\[3x - 4y = x - y + 4\]
Simplifying this involves:

• Moving similar terms to one side of the equation.
• Combining the terms appropriately.
• Reducing the equation to a simpler form, such as \(2x + 3y = 4\).

Understanding how to manipulate algebraic expressions is crucial for solving equations. It allows you to see the relationships between different terms and helps in expressing one variable in terms of others as required by the substitution method. Learning these skills will not only help in solving algebra problems but also provide the foundation for more advanced math topics.

Set Notation

Set notation is a mathematical way to represent a collection of elements. It's particularly useful in conveying the solutions of equations succinctly. When dealing with systems of equations, set notation is often used to express the solution set of the variables.

In the context of a system of equations, the solution set comprises all the possible pairs (or sets) of values that satisfy all equations within the system. For example, the solution to the system from the exercise is written in set notation as \(\{(3, -2)\}\). This means that the only x-y pair that satisfies both equations is \((x = 3, y = -2)\).

Using set notation helps in:

• Clearly identifying solutions that satisfy conditions.
• Communicating these solutions concisely.
• Handling solutions in advanced mathematical contexts.

Whether there is one solution, no solution, or an infinite number of solutions, set notation provides a compact way to convey this. It's a powerful tool that extends to various branches of mathematics and related fields.