Problem 61 - 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 \(x .\) $$3 x+5 x+4$$

Step-by-Step Solution

Answer

The value is 28.

Step 1: Understand the Expression

The given expression is made up of two terms that include the variable \(x\), which are \(3x\) and \(5x\), and a constant term, which is 4. Our task is to find the value of this expression when \(x = 3\).

Step 2: Simplify the Expression

We first simplify the expression by combining like terms. The terms \(3x\) and \(5x\) are like terms because they both have the variable \(x\). Adding these, we get: \(3x + 5x = 8x\). The expression simplifies to \(8x + 4\).

Step 3: Substitute the Value of x

Now that we have a simplified expression \(8x + 4\), we can substitute the value of \(x\) which is 3 into the expression. This gives us: \(8(3) + 4\).

Step 4: Calculate the Result

Multiply 8 by 3 to find the product: \(8 \times 3 = 24\). Then, add the constant term 4 to this product: \(24 + 4 = 28\).

Step 5: Write the Final Answer

The value of the expression \(3x + 5x + 4\) when \(x = 3\) is 28.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental concept in algebra. The goal here is to make complex mathematical expressions easier to handle and understand. To achieve this, you combine terms that are similar. In this context, "terms" are components of the expression separated by plus or minus signs.
For instance, in the expression \(3x + 5x + 4\), the terms \(3x\) and \(5x\) are considered like terms because they both contain the variable \(x\). Simplifying means combining these like terms:

• First, identify like terms. Here, \(3x\) and \(5x\) are like terms.
• Add the coefficients of these terms: \(3 + 5 = 8\).
• The simplified expression becomes \(8x + 4\).

This process reduces the expression's complexity, making it easier to evaluate or further manipulate.

Substitution

Substitution is a method used in algebra to replace a variable with a given value. This step often follows simplifying expressions and is crucial for finding specific numerical values of expressions. Imagine you have a simplified expression like \(8x + 4\), and you need to find its value when \(x = 3\). Substitution allows you to do just that by following these steps:

• Identify the variable in the expression: in this case, \(x\).
• Replace \(x\) with the value 3. This changes the expression from \(8x + 4\) to \(8(3) + 4\).
• Perform the arithmetic operation: calculate \(8 \times 3 = 24\).
• Add the constant term: \(24 + 4 = 28\).

Through substitution, you seamlessly find the expression's value and understand how particular values influence it.

Like Terms

Understanding like terms is essential in algebraic manipulation. Like terms are terms in an expression that have the same variable raised to the same power, which can be combined to simplify expressions.
Let's break it down with \(3x + 5x + 4\):

• The terms \(3x\) and \(5x\) are like terms because both have the variable \(x\) with the same exponent, which is implicitly 1.
• The coefficient of a term is the numerical part that multiplies the variable, such as 3 in \(3x\).
• Combine the coefficients of like terms: \(3 + 5 = 8\).
• This yields a single term \(8x\), which is simpler and combines the effects of both initial terms.

Like terms make simplifying expressions efficient and manageable, providing clarity and structure to algebraic equations.