Problem 30 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when a = 3 and c = 5. $$ a+c $$

Step-by-Step Solution

Answer

The evaluated expression is \(8\).

Step 1: Identify the Given Values

The given values are \(a = 3\) and \(c = 5\).

Step 2: Substitute the Given Values into the Expression

Replace the variables \(a\) and \(c\) in the expression \(a + c\) with the values provided. So, the expression becomes \(3 + 5\).

Step 3: Simplify the Expression

Solve \(3 + 5\) to get the simplified result.

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra that involves replacing variables with their corresponding values. This technique is particularly useful when evaluating variable expressions.

For example, given the expression, \(a + c\), and being told to evaluate it when \(a = 3\) and \(c = 5\), we start by replacing the variable \(a\) with 3 and \(c\) with 5. This is the essence of substitution and is illustrated as follows: \[ 3 + 5 \.\] After the substitution, you effectively have a numerical expression that is much simpler to deal with compared to the original algebraic expression.

Substitution is not limited to simple addition; it can be applied to all types of algebraic expressions that may include subtraction, multiplication, division, and even more complex operations. The key is to consistently replace each variable with its given value and then perform the arithmetic operations as required.

Simplifying Expressions

Once the substitution method has been applied, the next step in evaluating algebraic expressions is simplifying expressions. This essentially means performing the arithmetic operations to reduce the expression to its simplest form or numerical value.

In the context of our example, where the expression after substitution is \[ 3 + 5 \.\], we proceed by adding the numbers together. The process of simplification in this case is straightforward:
\[ 3 + 5 = 8 \.\]Simplifying expressions can become more complex with the inclusion of varied operation signs, exponents, or parentheses. However, the principle remains the same: combine like terms and execute operations according to the order of operations (PEMDAS/BODMAS). The goal is always to end with the most reduced form of the expression or a single numerical value.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations (such as addition, subtraction, multiplication, and division). Understanding algebraic expressions is key to mastering algebra, as they are the building blocks for forming equations and solving algebraic problems.

An expression like \(a + c\) is a simple algebraic expression consisting of two variables, \(a\) and \(c\), and one operation, addition. Contrasted with numeric expressions that contain only numbers, algebraic expressions can represent a variety of values since the variables can take on different numbers.

Exploring more complex expressions involves dealing with coefficients (numbers multiplied with variables), exponents (variables raised to a power), and operations within parentheses. The complexity of an algebraic expression doesn't change the basic approach to solving: always identify the variable values, substitute, and then simplify in accordance with arithmetic rules.