Problem 33 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The value of the expression for a = 3 and c = 5 is 15

Step 1: Substitution

Substitute the known values of a and c into the given expression. So, the given expression a * c becomes 3 * 5.

Step 2: Carry out the multiplication

Multiply the substituted values. 3 * 5 equals to 15.

Key Concepts

Substitution Method

When it comes to evaluating algebraic expressions, one of the fundamental techniques employed is the substitution method. The core principle behind this method is quite straightforward: replace the variables in an expression with their respective numerical values. This process transforms an abstract algebraic statement into a concrete arithmetic problem that can be solved using basic math operations.

For instance, consider the expression involving variables 'a' and 'c.' If the problem states that the value of 'a' is 3 and 'c' is 5, you would substitute these values directly into the expression. The expression a * c would become 3 * 5 upon substitution. This approach is universally applicable, no matter how complex the initial expression may be, making it an essential skill for any student tackling algebra.

Algebraic Expressions

At the heart of algebra lies the concept of algebraic expressions. These are mathematical phrases that can contain numbers, variables (like 'a' and 'c'), and arithmetic operations such as addition, subtraction, multiplication, and division. Unlike equations, algebraic expressions don't have an 'equals' sign; they are simply expressions that denote a quantity.

Let's focus on understanding these expressions, as they are the building blocks for more advanced algebraic principles. In an expression like a * c, 'a' and 'c' are variables that can take on different values, and the asterisk symbol '*' denotes multiplication. Learning how to manage and manipulate these expressions is crucial for progressing in algebra, as it sets the foundation for solving equations, graphing functions, and understanding mathematical models.

Multiplication of Variables

When dealing with variables, multiplication of variables is as simple as multiplying numbers. The process remains the same: you multiply the coefficients (numbers in front of the variables) and then apply the resulting product to the variables involved.

However, when variables are the same and are multiplied together (like a * a), we use exponents to simplify the expression (a^2), representing that 'a' is multiplied by itself. If our variables are different, as in the example of multiplying 'a' by 'c', we simply write them side by side (ac) to indicate that they have been multiplied together. It's important to remember that in algebra, when two variables are written together without a clear sign of operation between them, it implies multiplication.