Problem 65 - Algebra 1: Concepts and Skills

Question

EVALUATING EXPRESSIONS Evaluate the expression for the given value of the variable. (Lesson 1.1) \(12+x\) when \(x=4\)

Step-by-Step Solution

Answer

The solution is 16.

Step 1: Understand the Equation

We are given an equation \(12+x\), and we are asked to evaluate this expression when \(x = 4\). This means that we substitute the value of \(x\) into the equation where \(x\) appears.

Step 2: Substitute Value of x

Substitute \(x = 4\) into the equation. So the equation becomes \(12+4\).

Step 3: Solve the Equation

Add 12 and 4 together to get the answer, which is 16.

Key Concepts

Substitution

Substitution is a fundamental concept in mathematics used to simplify expressions and solve equations. When we talk about substituting in math, we're essentially replacing a variable with a given numerical value. It is a crucial skill because it allows you to evaluate expressions by simplifying them.

Consider the expression given, \(12 + x\). Here, \(x\) is a variable, which means it can take any value. When we substitute \(x = 4\), we replace every instance of \(x\) with the number 4. So, our expression \(12 + x\) becomes \(12 + 4\). Substitution helps to "personalize" the expression with the specific values we're interested in.

It's like filling in the blanks in a sentence. Performing substitution accurately is essential for further solving or simplifying mathematical problems. Once you understand how and where to substitute, this process becomes straightforward and intuitive.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They represent values that can vary depending on what the variables stand for.

In our example, the algebraic expression is \(12 + x\). It consists of:

• The constant number 12, which is fixed and doesn't change.
• The variable \(x\), which represents an unknown or changeable number.
• The plus sign, which indicates that addition is the operation involved.

Understanding algebraic expressions is key to evaluating them correctly. Each component of the expression serves a purpose, and together they define the mathematical statement.

Working with algebraic expressions allows us to generalize mathematical problems. This generalization is powerful as it permits both solving specific instances (by substitution) and finding general solutions or rules.

Basic Arithmetic Operations

Basic arithmetic operations are the foundation of most mathematical computations and include addition, subtraction, multiplication, and division. These operations allow us to combine, separate, increase, or decrease values.

In evaluating expressions, as in our given problem — \(12 + x\) when \(x = 4\) — we focus primarily on addition. After substituting \(x = 4\) into the expression, we calculate \(12 + 4\).

• Addition is simply combining two numbers to get a total.
• In this case, \(12 + 4\) results in 16.

Understanding basic arithmetic ensures that once you've substituted values into an expression, you can correctly simplify or solve it. These operations are universal, used not just in mathematics, but in everyday life, highlighting their fundamental importance.