Problem 48 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5.\) \(4 x\)

Step-by-Step Solution

Answer

The result of the expression is 4.

Step 1: Substitute the Values

Replace the variable \(x\) in the expression with the given value. The expression becomes \(4 \cdot 1\).

Step 2: Calculate the Expression

Multiply the numbers: \(4 \times 1 = 4\).

Key Concepts

Substitution Method

The substitution method is a powerful and straightforward technique to evaluate mathematical expressions. The idea is simple: replace each variable in the expression with its given numerical value. This allows you to convert the expression from one with variables to one that can be directly calculated. In this exercise, you were asked to evaluate an expression with the given value for the variable \(x\). The original expression was \(4x\). By substituting the value \(x=1\), the expression transforms into a more straightforward form: \(4 \cdot 1\).
Remember:

• Identify the variables that need substitution in the given expression.
• Replace each variable with the provided numerical values.
• Re-write the expression with these replacements so it can be simplified or calculated next.

Using the substitution method, you systematically make complex problems simpler, one step at a time.

Basic Multiplication

Understanding basic multiplication is crucial when moving from substitution to calculation. Multiplication is one of the fundamental operations in math. It tells us how many times to add a certain number. When you see a term like \(4x\), it means \(x\) needs to be multiplied by 4.
After substitution, the expression becomes \(4 \cdot 1\), illustrating basic multiplication in action. Here, the multiplication sign is a dot \(\cdot\), a common representation in algebra.

• When multiplying by 1, the number stays the same. For example, \(4 \times 1 = 4\).
• Apply multiplication consistently across any similar problems.
• This concept applies not only to numbers but also to algebraic terms.

Mastery of this fundamental operation sets the foundation for tackling more complex calculations and expressions.

Expression Simplification

Expression simplification makes mathematical expressions easier to read and solve. Once you've substituted variables and performed necessary operations, it's time to simplify.Simplification involves reducing an expression to its simplest form. For our exercise, after the substitution and multiplication, you ended with \(4\). This is already as simple as it gets, as we've removed any unnecessary components.Here’s how to simplify any expression:

• Perform operations like addition or multiplication as indicated.
• Combine like terms if possible - look for terms that are similar and combine them into one.
• Check if the expression can be further reduced (by removing parentheses or reducing factors).

By simplifying, you reduce errors and make the math clearer. With practice, recognizing opportunities to simplify will become second nature.