Problem 82 - Introductory Algebra for College Students

Question

In Exercises \(80-83,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(y=x-1\) and \(x=y+1\) are dependent.

Step-by-Step Solution

Answer

The statement that the equations \(y=x-1\) and \(x=y+1\) are dependent is true.

Step 1: Analyzing the First Equation

Starting with the first equation \(y=x-1\). The 'y' variable can be expressed as follows: \(y=x-1\).

Step 2: Analyzing the Second Equation

In the second equation \(x=y+1\), express 'x' in terms of 'y'. It gives: \(x=y+1\).

Step 3: Comparing the Equations

Compare the two equations. This comparison shows that \(y=x-1\) and \(x=y+1\) are rearrangements of each other. Hence, the two equations are dependent.

Key Concepts

System of Linear Equations

Understanding the concept of a system of linear equations is fundamental in algebra. In simple terms, a system of linear equations consists of two or more equations that share a set of variables and are considered together.

For instance, an example of such a system is where we have two equations, the first being y=x-1 and the second x=y+1. In this set, 'x' and 'y' are the common variables. The goal when working with these systems is to find a solution that satisfies all equations simultaneously.

A system of linear equations can have one solution, no solution, or infinitely many solutions. If the system has at least one solution, the equations are said to be 'consistent'; if not, they are 'inconsistent'. Within the consistent category, if there's exactly one solution, the equations are 'independent', and if there are infinitely many solutions, they are 'dependent'. The latter occurs when the equations are essentially the same line, which is the case in the exercise provided, indicating that all points on the line are shared solutions to both equations.

Solving Algebraic Equations

Solving algebraic equations is often about finding the value of variables that make the equation true. For a single linear equation, this usually involves isolating the variable on one side of the equation using basic algebraic operations: addition, subtraction, multiplication, and division.

In the context of our system, the equations y=x-1 and x=y+1 can be manipulated algebraically to solve for 'x' or 'y'. However, due to the nature of dependent equations, solving one will directly give us the form of the other, thus indicating a single line rather than a specific point of intersection.

It is crucial to apply inverse operations'step by step' to avoid mistakes. For example, to isolate 'y' in the first equation, we leave it as is, which becomes y=x-1. Similarly, to isolate 'x' in the second equation, we rewrite it as x=y+1. The logical flow of these steps must be clear and concise for the student to track each operation.

Analyzing Linear Relationships

Linear relationships represent connections between variables that can be graphed as straight lines on a coordinate plane. Analyzing these relationships involves understanding the slope and intercept of these lines, as they dictate the direction and location of the line, respectively.

In the case of dependent equations like y=x-1 and x=y+1, both lines would have the same slope and intercept, causing them to fall perfectly on top of each other when graphed. This reveals the 'dependent' nature of the system, as both equations describe the same line and thus, the same relationship between 'x' and 'y'.

Understanding that dependent lines have an infinite number of solutions is crucial. This infinite set represents all the points along the line where these relationships hold true. Encouraging students to graph these relationships can promote a deeper comprehension of how these lines interact, or in this case, coincide.