Problem 65 - 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 \(x .\) $$4 x+3+2 x+5$$

Step-by-Step Solution

Answer

The value of the expression when \(x=3\) is 26.

Step 1: Combine Like Terms

Examine the expression and combine the like terms. The given expression is: \[ 4x + 3 + 2x + 5 \]Combine the like terms for \(x\): \(4x + 2x = 6x\).Now, combine the constant terms: \(3 + 5 = 8\).Thus, the expression simplifies to: \[ 6x + 8 \]

Step 2: Substitute \(x = 3\)

Take the simplified expression from Step 1:\[ 6x + 8 \]and substitute \(x = 3\) into this expression:\[ 6(3) + 8 \]

Step 3: Perform the Multiplication

Compute the multiplication in the expression:\[ 6(3) = 18 \].Replace \(6(3)\) with \(18\), so the expression now reads:\[ 18 + 8 \]

Step 4: Add the Constant

Add the two constants together:\[ 18 + 8 = 26 \].

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental concept that makes solving expressions much simpler by replacing variables with known values. In algebraic terms, a variable is often a symbol like \( x \) that represents an unknown or variable number. When you know the value of this variable, you can substitute it directly into the expression.

Here's how you can think of it practically:

• Take the expression: \( 6x + 8 \).
• Substitute \( x = 3 \), which means wherever \( x \) appears, replace it with 3.
• The expression becomes \( 6(3) + 8 \).

This step simplifies the evaluation process, turning an abstract algebraic problem into a straightforward arithmetic task. Substitution helps in focusing on numerical calculations rather than dealing with abstract symbols.

Combining Like Terms

In algebra, combining like terms is an essential skill that helps in simplifying expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression \( 4x + 3 + 2x + 5 \), \( 4x \) and \( 2x \) are like terms because both are multiplied by \( x \).

The process is simple:

• Identify terms with the same variable: \( 4x \) and \( 2x \).
• Add their coefficients: \( 4 + 2 = 6 \).
• The expression part becomes \( 6x \).

Similarly, combining constant terms like \( 3 \) and \( 5 \) results in \( 8 \). Combining like terms reduces complexity, creating a more compact expression that's easier to work with:

\( 6x + 8 \).

Simplifying Expressions

Simplifying expressions involves reducing complexity while maintaining equivalence. This often involves both combining like terms and performing any basic arithmetic operations needed post-substitution.

The steps generally include:

• First, combine like terms to make the expression concise.
• Next, substitute any known values for the variables (as seen with substituting \( x = 3 \)).
• Finally, carry out operations like addition, subtraction, multiplication, and division.

For example, after substituting \( x = 3 \) into \( 6x + 8 \), it becomes \( 6(3) + 8 = 18 + 8 \).

Perform the operations step-by-step, resulting in the final simplified answer: 26. This simplification makes the problem more approachable and the answer clearer.