Problem 73 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each algebraic expression for the given value of the variable. $$x^{2}+5 x ; x=3$$

Step-by-Step Solution

Answer

The evaluated result is 24

Step 1: Substitute the given value into the equation

To start, substitute the given x value into the algebraic expression \(x^{2}+5 x\) which would give \(3^{2}+5*3\)

Step 2: Simplify the power

Next, simplify \(3^{2}\) to give \(9\)

Step 3: Perform the multiplication

Now, perform the multiplication \(5*3\) to give \(15\)

Step 4: Final calculation

Add the results from Step 2 and 3 together to get \(9 + 15 = 24\)

Key Concepts

Substitution Method

When evaluating algebraic expressions, the substitution method is an essential technique that involves replacing variables with their given numerical values. In the context of our exercise, where the algebraic expression is \(x^{2}+5x\) and the value of x is given as 3, we applied this method in the first step.

To execute the substitution effectively, you simply replace each instance of the variable with the number it represents, ensuring that you follow proper arithmetic rules, such as maintaining the order of operations. This process transforms the abstract algebraic equation into a concrete arithmetic one that is straightforward to resolve.

Remember, the substitution method is the foundational step in the process of evaluating expressions and is widely used across various fields of math and science.

Simplifying Expressions

Moving past substitution, simplifying expressions is the necessary next step to find your answer. This process involves performing arithmetic operations in the correct sequence, often guided by the order of operations (PEMDAS/BODMAS).

In the given exercise, after substituting 3 for x, we simplified the expression starting with the power operation: \(3^{2}\), which results in the number 9. Simplifying expressions can include combining like terms, reducing fractions, and carrying out exponentiation, among other operations. An expression is fully simplified only when no further arithmetic operations can be performed.

It's imperative for students to master the simplification process to tackle more complex equations efficiently and to adapt these skills for solving real-world problems.

Arithmetic Operations

Arithmetic operations, such as addition, subtraction, multiplication, and division, are the building blocks of evaluating algebraic expressions. In our solution, after simplifying the power, we performed a multiplication operation: \(5 \times 3\), yielding 15.

The final step involves adding our simplified components together, in this instance, 9 (from the power of 3) plus 15 (from the multiplication), to get the final answer of 24. Understanding and correctly applying these operations is critical for successfully evaluating expressions. Students are encouraged to practice these basics to ensure they are second nature, which is fundamental for progress into more advanced mathematical concepts.