Problem 66 - Prealgebra

Question

Find the value of each of the following expressions when \(x=3 .\) You may substitute 3 for \(x\) in each expression the way it is written, or you may simplify each expression first and then substitute 3 for \(x .\) $$7 x+6+2 x+9$$

Step-by-Step Solution

Answer

The value of the expression when \(x = 3\) is 42.

Step 1: Understand the Expression

We are given the mathematical expression: \(7x + 6 + 2x + 9\). Our task is to find its value by substituting \(x = 3\). It is a combination of terms involving \(x\) as well as constants.

Step 2: Simplify the Expression

First, combine like terms. We have \(7x\) and \(2x\) which are like terms. Add them together to get \((7 + 2)x = 9x\). The expression simplifies to \(9x + 6 + 9\).

Step 3: Combine Constant Terms

Add the constant terms \(6\) and \(9\). This gives us \(15\). Now our expression is \(9x + 15\).

Step 4: Substitute \(x = 3\) into the Simplified Expression

Replace \(x\) with \(3\) in \(9x + 15\). This gives us \(9(3) + 15\).

Step 5: Calculate the Expression

Multiply \(9\) by \(3\) to get \(27\). Then add \(27\) and \(15\) to get \(42\).

Step 6: Final Answer

The value of the expression when \(x = 3\) is \(42\).

Key Concepts

Substitution

Substitution is the process of replacing variables in an algebraic expression with specific values or numbers. In this exercise, we substitute the variable \(x\) with the number 3 in the algebraic expression \(7x + 6 + 2x + 9\).

• The first step is to identify where substitution is needed. Here, we focus on the terms \(7x\) and \(2x\), as they're the ones carrying the variable \(x\).
• Once we simplify the expression, we substitute \(x = 3\). So the expression \(9x + 15\) becomes \(9(3) + 15\).
• By substituting, we turn the algebraic expression into an arithmetic problem that can easily be solved to get the final value.

The main goal of substitution is to simplify expressions using known values, making it a powerful technique in algebra.

Simplification

Simplification is all about making an algebraic expression easier to work with by reducing it to its simplest form. In the given expression \(7x + 6 + 2x + 9\), we can simplify by combining like terms.

• Like terms are terms that contain the same variable, they can be added together. Here, \(7x\) and \(2x\) are like terms.
• We add them together to get \(9x\), which is a more compact form of the variable terms.
• In addition, constants \(6\) and \(9\) are added together to form \(15\).

After simplification, the expression becomes \(9x + 15\), which is simpler and prepares it for further operations like substitution.

Like Terms

Like terms are a fundamental concept in algebra. They are terms that contain the same variables raised to the same power.

• In our exercise, \(7x\) and \(2x\) are like terms because both terms contain the variable \(x\) with the same exponent, which is 1 in both cases.
• We combine like terms by simply adding or subtracting the coefficients (the numerical parts). For example, \(7x + 2x\) simplifies to \(9x\).
• This concept is crucial because it allows us to simplify expressions, making calculations more straightforward and manageable.

Identifying and working with like terms is an essential skill in prealgebra and helps lay the foundation for more advanced mathematical operations.

Prealgebra

Prealgebra is foundational mathematics, and understanding its concepts is crucial for tackling more advanced topics like algebra, calculus, and beyond. It introduces basic arithmetic operations and lays the groundwork for understanding algebraic expressions.

• In prealgebra, students learn to manipulate numbers and variables, preparing them for equations and expressions.
• In this context, we evaluate expressions by substituting numerical values and simplifying them using operations like addition and multiplication.
• Substitution and simplification, as illustrated in this exercise, form part of the core competencies you gain in prealgebra, enhancing critical thinking and problem-solving skills.

Mastering prealgebra sets the stage for confidence and success in higher-level mathematics courses.