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Chapter 0: Problem 63

Evaluate each algebraic expression for \(x=2\) and \(y=-5\). $$|x|+|y|$$

Short Answer

Expert verified
The value of the expression \(|x| + |y|\) for \(x=2\) and \(y=-5\) is 7.

Step by step solution

01

Understand and Apply the Absolute Value Function

First, recall the basic rule for absolute value: |x| is x if x is positive or zero, and -x if x is negative. Follow the formula: \( |a| = \begin{cases} a & \text{if } a \geq 0 \ -a & \text{if } a < 0 \end{cases}\). Now, substitute the given values into the expression.
02

Substitute the Values for x and y

Replace x and y in the given expression. So, \(|x| + |y|\) becomes \(|2| + |-5|\). Apply the absolute value operation on these numbers.
03

Simplify the Expression

After application of the absolute value operation, our expression simplifies to 2 + 5 which equals 7. So, value of the expression \(|x| + |y|\) for \(x=2\) and \(y=-5\) is 7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution in Algebraic Expressions
When you have an algebraic expression, it often contains variables like \(x\) and \(y\). Substitution means replacing these variables with numbers to evaluate the expression. This process allows you to determine what the expression equals when specific values are assigned.
For instance, in our expression \(|x| + |y|\), \(x\) and \(y\) can represent any number.
  • In our example, we start with \(x = 2\) and \(y = -5\).
  • We replace \(x\) with 2 and \(y\) with -5 to evaluate the expression.
Substitution is a crucial first step because it gives you specific values to work with, leading to the next phase of solving or simplifying the expression. Always ensure you substitute correctly to avoid errors later in the process.
Simplifying Expressions
Once substitution has been performed in an algebraic expression, you often need to simplify the expression to find an answer. Simplifying makes the expression easier to understand and solve.
In the case of absolute value expressions, simplifying involves applying the definitions of absolute value:
  • The absolute value of a positive number or zero is the number itself.
  • The absolute value of a negative number is its opposite (positive version).
Let's see this in action:
For \(|x| + |y|\) with \(x=2\) and \(y=-5\):
  • Replace \(|x|\) with \(2\) because the absolute value of 2 is 2.
  • Replace \(|y|\) with \(5\) because the absolute value of -5 is 5.
This gives us the simplified expression \(2 + 5\). The simplification showcases the clarity derived after replacing and reevaluating the values within an expression.
Algebraic Evaluation
Algebraic evaluation is the final step where you compute the numeric value of the expression after substitution and simplification. It involves performing arithmetic operations such as addition, subtraction, multiplication, or division.
In our problem, the final step is straightforward: add the absolute values we've found.
  • To evaluate \(2 + 5\), simply perform the addition to get 7.
Thus, the evaluation of \(|x| + |y|\) for \(x = 2\) and \(y = -5\) results in 7.
This step confirms the result and brings closure to the process, offering a precise answer that fulfills the expression's requirements. It's important to ensure that all previous steps were done correctly to guarantee this value is accurate.

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Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

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What is a linear equation in one variable? Give an example of this type of equation.
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