Problem 11 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(4 x\)

Step-by-Step Solution

Answer

When \(x = 4\), the expression is \(16\). When \(x = 6\), the expression is \(24\).

Step 1: Substitute \(x = 4\) into the expression

To find the value of the expression when \(x = 4\), substitute \(4\) for \(x\) in the expression \(4x\). This gives us: \[4 \times 4\]

Step 2: Calculate the expression for \(x = 4\)

Multiply \(4\) by \(4\): \[4 \times 4 = 16\]

Step 3: Substitute \(x = 6\) into the expression

To find the value of the expression when \(x = 6\), substitute \(6\) for \(x\) in the expression \(4x\). This gives us: \[4 \times 6\]

Step 4: Calculate the expression for \(x = 6\)

Multiply \(4\) by \(6\): \[4 \times 6 = 24\]

Key Concepts

Substitution

Substitution is a key concept in evaluating algebraic expressions. It involves replacing a variable with a specific number. For example, if you have the expression \(4x\) and you know that \(x = 4\), you can replace \(x\) with \(4\). This is done to simplify the expression and make it possible to calculate a specific value.
Let's look at an example:
Given the expression \(4x\) and that \(x = 4\), you substitute \(4\) for \(x\) :
\(4x \rightarrow 4 \times 4\)
This makes the algebraic expression easier to solve.
Common steps involved in substitution include:

• Identifying the variable in the expression.
• Substituting the variable with the given number.
• Simplifying the resultant expression through basic arithmetic operations.

For instance, substituting \(x = 6\) in \(4x\) would look like \(4 \times 6\).

Multiplication

Multiplication is one of the fundamental arithmetic operations and is crucial when evaluating expressions like \(4x\). Once you substitute the variable with a specific number, as learned previously, the next step is often to multiply.
For example, after substituting \(x = 4\) into \(4x\), we get:
\(4 \times 4\)
This is a straightforward multiplication problem, which you can solve easily:
\[4 \times 4 = 16\]
Similarly, if \(x = 6\):
\(4 \times 6 = 24\)
Multiplication helps us to find the exact value of the expression after substitution.
It involves repeated addition and is often represented by the symbol ×. Key steps in multiplication include:

• Identifying the two numbers to be multiplied (the coefficient and the substituted value).
• Multiplying them together to get the product.

Understanding these basics of multiplication will help you simplify and solve more complex algebraic equations.

Variable Evaluation

Variable evaluation is the process of calculating the exact value of an algebraic expression based on the given variables. It combines both substitution and multiplication. Here’s an in-depth look at how to evaluate an expression like \(4x\) for different values of \(x\):

• Start with the expression \(4x\).
• Substitute the variable (\(x\)) with the given value. Suppose we have two different cases: \(x = 4\) and \(x = 6\).
• For \(x = 4\):
  Replace \(x\) with 4 to get \(4 \times 4\).
• Multiply to get \(16\).
• For \(x = 6\):
  Replace \(x\) with 6 to get \(4 \times 6\).
• Multiply to get \(24\).

This process helps in transforming the algebraic expression into a numerical value. Using this evaluated number, one can solve bigger equations and understand relationships between variables. Remember, the key stages of variable evaluation include:
Identifying the variable, substituting it with a given number, and performing arithmetic operations.