Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=4\). $$5+3 x$$

Step-by-Step Solution

Answer

The value of the expression \(5 + 3x\) for \(x = 4\) is 17.

Step 1: Substitute the value of \(x\) into the expression

Replace \(x\) in the expression \(5 + 3x\) with the given value, which is 4. So it becomes \(5 + 3 \cdot 4\).

Step 2: Simplify the expression

Now simplify the expression by performing the multiplication before the addition according to the order of operations (BIDMAS/BODMAS). Firstly, calculate \(3 \cdot 4\), which gives 12. So the expression now becomes \(5 + 12\). Secondly, add 5 to 12 which gives 17.

Key Concepts

Substitution

Substitution in algebra involves replacing variables with their corresponding values. It is a critical step when dealing with algebraic expressions. For instance, in our problem, you're tasked with evaluating the expression \(5 + 3x\) when \(x = 4\). This simply means that wherever you see \(x\), you plug in the number 4.
This transforms the expression from \(5 + 3x\) to \(5 + 3 \cdot 4\). Remember, substitute the value provided to replace any instance of the variable. Doing so sets the stage for further simplification and ensures the expression's accuracy.
When substituting, write out the new expression clearly. Keep track of the mathematical operations you need to perform next. This deliberate process prevents errors and misunderstandings. By mastering substitution, you're tackling algebraic problems with a strong foundation.

Order of Operations

Order of Operations is a set of rules that dictates the correct sequence to evaluate parts of a math expression. It eliminates ambiguity and ensures you get the right answer. In math, we often refer to this rule with the acronym BIDMAS or BODMAS, which stands for Brackets, Indices (or Orders), Division, Multiplication, Addition, and Subtraction.
In our expression \(5 + 3 \cdot 4\), we need to remember that multiplication comes before addition according to these rules. This means that before adding 5, you first multiply 3 by 4. So you calculate \(3 \cdot 4 = 12\). Then, you add 5, resulting in 17.
Following the order of operations guarantees that everyone evaluates expressions consistently and arrives at the same answers. As you work through algebra, always think about BIDMAS/BODMAS to guide your calculations. Practice with more examples to become confident.

Simplification

Simplification involves reducing an expression to its simplest form. This process makes expressions easier to understand and use. For the expression \(5 + 3 \cdot 4\), simplification means performing the calculations to arrive at a single numeric value.
Starting with \(5 + 12\) after the multiplication, we then perform the addition. It gives us the final simplified result of 17. Simplification is about taking all these necessary steps to transform a complex expression into something more workable.
When simplifying:

• Perform operations in the correct order.
• Combine like terms where possible.
• Keep your work neat to avoid mistakes.

Mastering simplification is a crucial skill that allows you to handle more complicated algebraic expressions with ease. By breaking expressions down step-by-step, you can simplify them effectively.