Problem 5 - Algebra 1: Concepts and Skills

Question

Complete: You _______an expression by substituting numbers for variables and simplifying. The resulting number is called the______ of the expression.

Step-by-Step Solution

Answer

The completed sentences are: 'You evaluate an expression by substituting numbers for variables and simplifying. The resulting number is called the value of the expression.'

Step 1: Identify the first blank

When you substitute numbers for variables in an expression and simplify, you are evaluating the expression.

Step 2: Identify the second blank

The resulting number is called the value of the expression.

Step 3: Complete sentence

You evaluate an expression by substituting numbers for variables and simplifying. The resulting number is called the value of the expression.

Key Concepts

Substitution

Substitution is a fundamental step in evaluating expressions. When faced with a mathematical expression that includes variables, such as \(x\), \(y\), or \(z\), substitution involves replacing these variables with given numerical values. This process allows for the conversion of a general expression into a specific numerical form.

• Identify the variables within the expression.
• Replace each variable with its corresponding numerical value.
• Ensure all parts of the expression reflect this change.

For example, in the expression \(3x + 2\), if \(x = 4\), substituting \(x\) gives \(3(4) + 2\). This is the first step toward simplifying the expression and finding its value.

Simplifying Expressions

Simplifying expressions is the next crucial stage after substitution. Once the variables are replaced with numbers, the expression may still be complex. Simplification eases the problem-solving process and ensures the expression is presented in its most straightforward form.

• Start by performing arithmetic operations according to the order of operations: parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).
• Reduce fractions if required.
• Combine like terms, where applicable, to lessen the complexity.

Continuing with our previous example: \(3(4) + 2\) simplifies to \(12 + 2\), and further simplifies to \(14\). The goal is to make the process of finding the value of expressions as uncomplicated as possible.

Value of an Expression

Once you've substituted and simplified an expression, the resulting number is what we call the 'value of the expression'. This value is significant as it gives a precise outcome that can be used in further calculations or real-world scenarios.

• The 'value' represents a single numerical answer derived from the expression.
• It is obtained once all operations are complete and all terms have been simplified.
• Understanding the value helps in verifying the outcomes and making decisions based on the findings.

In the given context, the value of our example, \(3x + 2\) when \(x = 4\), is \(14\). This entire process, from substitution to simplification, leads us to clearly understand what the expression evaluates to.