Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The value of the expression \(3+5x\) for \(x=4\) is \(23\)

Step 1: Express the given problem

The problem to solve is \(3+5x\) for \(x=4\). This means replacing every occurrence of \(x\) in the expression with the value given, 4.

Step 2: Substitute the value of \(x\)

Substitute \(x=4\) into the given expression, we get: \(3+5(4)\). This simplification results in \(3 + 20\).

Step 3: Simplify the expression

Combining like terms, we get the final result which is \(23\).

Key Concepts

Expression Evaluation

Understanding expression evaluation is crucial when working with algebraic expressions. In algebra, an expression is a combination of numbers, variables, and mathematical operators. To evaluate an expression, you determine its value by substituting given values for its variables. In this specific exercise, you are given the expression \(3+5x\) with a specified value for \(x\).

To evaluate this expression:

• Identify the variable in the expression, which in this case is \(x\).
• Substitute this variable with the given value.
• Proceed with arithmetic operations to find the expression's value.

This process of evaluating expressions is foundational to learning more complex algebraic manipulations in college mathematics.

Substitution Method

The substitution method is a technique used not only in solving equations but also in expression evaluation. It involves replacing each variable in the expression with a given or known value. This method simplifies the process, letting you work with numbers instead of variables.

In the example provided, substitution occurs in a few simple steps:

• Begin with the expression \(3+5x\).
• Replace \(x\) with the given value, which is \(4\) in this scenario.
• The expression transforms into \(3+5 \times 4\), making it simple to evaluate.

This method allows algebraic expressions to be transformed into numerical ones, making calculation straightforward.

Simplification

Simplification is a key step after substitution that enables you to express an equation or expression in its simplest form. The aim is to combine and reduce terms in order to derive an easy-to-understand result.

In our example:

• Once \(x\) is replaced with \(4\), the expression \(3+5(4)\) emerges.
• Follow basic arithmetic rules to multiply and add: \(3 + 20\).
• Combine like terms to simplify: this results in a simple, final value of \(23\).

Simplification can reduce errors and make expressions less complicated, often yielding quicker and more accurate results.

College Mathematics

College mathematics builds upon fundamental concepts such as expression evaluation and simplification. Mastery of these basics lays the ground for tackling more advanced topics.

While high school algebra often introduces the foundations, college-level work requires a deeper understanding of processes like substitution and simplification. Applying these skills efficiently helps in calculus, linear algebra, and other higher-level math courses:

• Problem-solving: These methods are essential in solving real-world mathematical problems.
• Logical reasoning: They develop critical thinking skills by applying logical steps to reach a solution.
• Analysis: Evaluating expressions accurately is crucial in analyzing functions and modeling data.

Thus, the concepts illustrated in the example are not just academic exercises, but essential building blocks for future mathematical endeavors.