Problem 28 - 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. $$\left\\{\begin{array}{l} 3 x=4 y+1 \\ 4 x+3 y=1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set to the system of equations is \[\{(1/7, -1/7)\}\]

Step 1: Multiply the equations

In order to get rid of one of the variables, we first need to make the coefficients of x or y same in both equations. For this set of equations, let's multiply first equation by 4 and second equation by 3. That yields: \[12x = 16y + 4\] and \[12x + 9y = 3\]

Step 2: Subtract the equations

Subtract second equation from the first to get: (12x - 12x) = (16y - 9y) + (4 - 3). This simplifies to 0 = 7y + 1. Solving this equation for 'y' gives the result y = -1/7

Step 3: Substitute y = -1/7 in the first equation

Replacing 'y' by -1/7 in the first equation: 3x = 4*(-1/7) + 1 yields 3x = -4/7 + 1, which simplifies to 3x = 3/7. By solving this equation for x, we receive x = 1/7.

Step 4: Express the solution as a set

The solution set is {x=1/7, y=-1/7} expressed in set notation is: \[\{(1/7, -1/7)\}\]

Key Concepts

Systems of Equations

Understanding systems of equations is a fundamental part of algebra. A system of equations consists of two or more equations with the same set of variables. Our goal is to find the values for these variables that satisfy all the equations in the system simultaneously. In the problem given, we deal with two linear equations: - \(3x = 4y + 1\) - \(4x + 3y = 1\) We need to determine the values of \(x\) and \(y\) that make both equations true at the same time. This often involves using techniques such as the addition method, substitution, or graphing. The addition (or elimination) method is a practical strategy for solving systems, particularly when you want to eliminate one of the variables to make solving simpler.

Set Notation

Set notation is a formal way to express the solution of a system of equations. It provides a clear presentation of the solutions, which includes each variable accounted for in a structured manner. For a solution set, such as presented in this problem, it looks like this: \[ \{(1/7, -1/7)\} \]This indicates that there is one solution for the system where \(x = 1/7\) and \(y = -1/7\). Using braces \( \{\} \), we specify that these values, together, form the solution set. Depending on the system's nature, solution sets can be empty, finite like this one, or infinite, including all possible pairs that satisfy the equations. This notation helps to compactly represent our results in mathematical computations.

Algebraic Solution Steps

Solving systems of equations with the addition method involves several clear steps to reach the solution. Here is how it's done:

• **Step 1: Equalize Coefficients** –– Choose one of the variables to eliminate from both equations. In the provided solution, we multiplied the first equation by 4 and the second by 3 to have matching coefficients for \(x\) (both \(12x\)).
• **Step 2: Subtract the Equations** –– Subtract one equation from the other, which helps eliminate \(x\) and solve for \(y\). Here, we ended with the equation \(0 = 7y + 1\), leading to \(y = -1/7\).
• **Step 3: Substitute Back** –– Use the value of \(y\) to find \(x\). Substitute \(y = -1/7\) back into one of the original equations. We used the first equation to solve for \(x\), resulting in \(x = 1/7\).
• **Step 4: Solution Set** –– Express the found values in set notation, making it clear that \((x, y) = (1/7, -1/7)\) is the solution.

These steps guide the process, ensuring we find the solution systematically, and they can be applied to a wide range of similar algebraic problems.