Problem 44 - Algebra for College Students

Question

\(|x+y|=1\)

Step-by-Step Solution

Answer

The solutions are two equations: \(x+y=1\) and \(x+y=-1\).

Step 1: Understanding Absolute Value

The absolute value of a number or expression is its distance from zero on a number line, and it is always non-negative. Thus, \(|x+y|=1\) means that the expression \(x+y\) is either 1 or -1.

Step 2: Set Up the Equations

From the absolute value expression, we deduce two possible equations: 1. \(x+y = 1\) 2. \(x+y = -1\).

Step 3: Solve for One of the Variables

For each equation, you can express one variable in terms of the other. For example:1. From \(x+y=1\): \(x = 1-y\)2. From \(x+y=-1\): \(x = -1-y\).

Step 4: Consider Multiple Solutions

In absolute value problems, both scenarios need to be considered unless more information is provided to find specific values. Each equation represents a line if you were to graph them.

Key Concepts

Number Line

The number line is a basic but powerful tool in mathematics for visualizing numbers and understanding their relationships.
When dealing with absolute value equations, it helps to think about the distance of numbers from zero. The term "absolute value" refers to the non-negative distance of a number from zero on this line.

• For example, both 3 and -3 have an absolute value of 3, because they are both three units away from zero.
• This concept is crucial because it simplifies the process of visualizing solutions to absolute value equations.

By plotting numbers on a number line, we gain a visual understanding of how expressions, like \(x+y\), satisfy conditions such as \(|x+y| = 1|\).
The possible solutions, \(x+y=1\) and \(x+y=-1\), indicate that the expression can lie at those two distinct points from zero, further showing the utility of the number line in these problems.

Expression Solving

Expression solving involves computing unknown variables by manipulating mathematical expressions. It's crucial to understand its use in solving absolute value equations like \(|x+y|=1\).
This equation symbolizes that the composite expression \(x+y\) can yield either 1 or -1. Here's how this plays out:

• First, identify possible scenarios for \(x+y\): either the expression equals 1 or it equals -1, based on the absolute value principle that accounts for both positive and negative solutions.
• Next, setup corresponding equations: one where \(x+y = 1\) and another with \(x+y = -1\).

For each of these equations, solving for a variable involves isolating it on one side.
For instance, rewrite \(x+y=1\) as \(x = 1-y\), or from \(x+y=-1\), derive \(x = -1-y\).
This step-by-step breakdown is fundamental in formulating all potential solutions.

Graphing Linear Equations

Graphing linear equations is an effective strategy for visually analyzing solutions to equations like those derived from absolute value problems. In this context, each scenario from \(|x+y|=1\) can be represented as a line on a Cartesian plane.
The given equations \(x+y=1\) and \(x+y=-1\) correspond to two linear graphs:

• The graph of \(x+y=1\) is a line with a positive slope intersecting the y-axis at 1.
• The line \(x+y=-1\) follows a similar slope but intersects the y-axis at -1.

These are straight lines because the relationships between x and y are linear. Graphically displaying these lines helps in understanding the different sets of values satisfying the equation.
Each point on these lines represents a valid solution where \(x+y\) is either 1 or -1, reaffirming our equation-solving process. This visual method ensures all possible solutions are considered, aiding in a complete understanding.