Problem 62 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \((L e s s o n \quad 1.1)\). $$9t\quad when\quad t=7$$

Step-by-Step Solution

Answer

The value of the expression \(9t\) when \(t=7\) is 63.

Step 1: Identify the Expression and the Value of the Variable

The expression given in the exercise is \(9t\). The value provided for the variable \(t\) is 7.

Step 2: Substitute the Value of the Variable into the Expression

The variable \(t\) in the expression is replaced with the given value, 7. This results in the new expression \(9 * 7\).

Step 3: Simplify the Expression

Multiplying 9 and 7, the expression simplifies to 63

Key Concepts

Substitution Method in Algebra

The substitution method in algebra is a fundamental skill that involves replacing variables with their numerical values. This approach is crucial for evaluating algebraic expressions and solving equations. For example, if you're given an expression such as \(9t\) and told that \(t = 7\), you apply the substitution method by taking the concrete value of \(t\) and replacing the variable with it.

In practice, this would look like \(9t\) becoming \(9 \times 7\) upon substitution. It's essentially like swapping out a placeholder for a real number, allowing you to perform arithmetic to simplify the expression. The substitution method is not only used in elementary algebra but also in more complex subjects like calculus, where it plays a role in integration and solving differential equations. Once the substitution is made, the next step is simplifying the expression to find the solution.

Simplifying Expressions

Simplifying expressions is the process of breaking down an algebraic expression into the simplest form possible. This often involves combining like terms, using the distributive property, and performing basic arithmetic operations. After substituting the variable with a known value, the simplification process starts, as in our example with \(9t\) when \(t = 7\).

The multiplication of 9 and 7 condenses the expression to a single number, 63. The goal of simplification is to make the expression as clear and concise as possible. It also prepares the expression for further operations, such as solving for variables in equations or evaluating functions. Learning to simplify expressions helps students understand algebraic structure better and develops their mathematical thinking.

Variable Manipulation

Variable manipulation involves rearranging and transforming algebraic expressions to solve for variables or to make calculations easier. It includes skills such as factoring, expanding, and using inverse operations to isolate variables. It is essential when trying to find the value of one or more variables in an equation or when you're working to understand how changing a variable impacts the expression.

In our example with \(9t\), variable manipulation is quite straightforward as we only have one multiplication operation to perform. However, in more complex expressions, you might need additional steps like adding, subtracting, or using exponents. Mastering variable manipulation empowers students to tackle a wide variety of algebraic challenges. Effective manipulation builds the foundation for higher-level math and is applicable in many real-world scenarios.