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Chapter 1: Problem 34

Evaluate each expression if \(x=-2, y=5,\) and \(z=-3 .\) $$x-z$$

Short Answer

Expert verified
The value of the expression \(x - z\) where \(x = -2\) and \(z = -3\) is 1.

Step by step solution

01

Identify the Given Variables

The variables are given as \(x=-2\), \(y=5\), and \(z=-3\). The expression to be evaluated is \(x-z\).
02

Substitute the Variables

Substitute \(x\) with \(-2\) and \(z\) with \(-3\) in the given expression. So, \(x-z\) becomes \(-2 - (-3)\).
03

Simplify the Expression

Since subtracting a negative number is the same as adding its absolute value, \(-2 - (-3)\) will simplify to \(-2 + 3\).

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Key Concepts

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

Evaluating Expressions
Evaluating expressions in algebra involves simplifying an expression or finding its numerical value based on given variable values. Imagine expressions as instructions for calculations. When you evaluate these expressions, you're essentially performing the operations they describe. For
  1. Start by identifying the expression to be evaluated. In our example, we have the expression \(x - z\).
  2. Note the values given for each variable. Here, \(x = -2\) and \(z = -3\).
  3. Substitute the given values into the expression to replace the variables. Once substitution is complete, calculate the expression to find the answer.
Evaluating expressions helps in understanding how changing variable values affect the result of a mathematical expression.
Substitution Method
The substitution method is a key technique in algebra used to replace variables with their known values so that an expression can be solved. This technique is straightforward:
  • Identify the variables in the expression and their corresponding values given in the problem statement.
  • Replace each variable in the expression with its specific value. For example, replacing \(x\) in the expression \(x - z\) with \(-2\) results in \(-2 - z\).
  • Substitute all variables in the expression to transform it into a purely numerical statement that can be easily calculated.
This method simplifies complex expressions into calculable forms, making it easier for you to arrive at a solution.
Variable Manipulation
Variable manipulation involves performing mathematical operations on variables to simplify an expression or solve an equation. In our exercise, we start with the expression \(x - z\):
  • First, substitute the given values for \(x\) and \(z\).
  • Next, recognize the operation involved. Subtracting \(-3\) is equivalent to adding \(3\), due to the rule of double negatives.
  • Now you can simplify \(-2 - (-3)\) to \(-2 + 3\), which equals \(1\).
Understanding how to manipulate variables and operations allows you to break down and solve expressions efficiently. This skill is crucial for more advanced algebraic problems down the line.

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Most popular questions from this chapter

A woman earns \(\$ 48,712.32\) in taxable income. She must pay \(15 \%\) tax on the first 823,000 and \(28 \%\) on the rest. In addition, she must pay a Social Security tax of \(15.4 \%\) on the total amount. How much tax will she need to pay?

A high-efficiency home heating system can be installed for \(\$ 4,170,\) with an average monthly heating bill of \(\$ 57.50 .\) A regular furnace can be installed for \(\$ 1,730,\) but monthly heating bills average \(\$ 107.75 .\) After three years, which system has cost more altogether?

In electronic data processing, the process of sorting records into sequential order is a common task. One sorting technique, called a selection sort, requires \(C\) comparisons to sort \(N\) records, where \(C\) and \(N\) are related by the formula $$ C=\frac{N(N-1)}{2} $$ How many comparisons are necessary to sort 10,000 records?

Let \(x=8, y=4,\) and \(z=2 .\) Write each phrase as an algebraic expression, and evaluate it. Assume that no denominators are \(0 .\) What factors are common to the second and third terms? Consider the algebraic expression \(3 x y+y+25 x y z\).

Which property of real numbers justifies each statement? $$(3+x)+y=3+(x+y)$$
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