Problem 40 - Elementary Algebra Within Reach

Question

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(64-16 t^{2}\) (a) \(t=2\) (b) \(t=3\)

Step-by-Step Solution

Answer

For \(t=2\), the evaluated algebraic expression is 0. For \(t=3\), the evaluated algebraic expression is -80.

Step 1: Evaluation for t = 2

First, we substitute \(t = 2\) into the given expression: \(64-16(2^{2}) = 64-64 = 0\)

Step 2: Evaluation for t = 3

Now, we substitute \(t = 3\) into the given expression: \(64-16(3^{2}) = 64-144 = -80\)

Key Concepts

Substitution

Substitution in math is like filling in the blanks with specific numbers. Imagine you have an expression, which is a mix of numbers and letters (like a recipe). In this case, the letter is a placeholder for a number that we can change. This process of replacing a variable with a number is called substitution. Let's say we have the expression \(64-16t^2\). To evaluate it:

• First, we take the given value for the variable \(t\) from our problem.
• Then, we "substitute" this value into the expression, replacing every \(t\) with that value.
• Finally, we solve the expression with the numbers filled in.

For example, if \(t=2\), you would replace \(t\) with \(2\), and solve \(64-16(2^2)\). It's just simple arithmetic after that!
This helps clarify what each expression equals when we use a specific number.

Algebraic Expressions

Algebraic expressions are like the sentences in the language of mathematics. They combine numbers and variables using mathematical operations such as addition, subtraction, multiplication, and division. Think of an algebraic expression as a math phrase that describes a certain situation or rule. For example, \(64 - 16t^2\) is an expression because it shows a rule relating \(t\) to some number we want to find.

Here's how they work:

• The numbers in the expression are constants; they don't change.
• The variables are the letters that represent numbers, and they can change.
• The operations show us what to do with the numbers and variables.

Algebraic expressions are powerful because they let us find results for different scenarios by using substitution. This way, they help us understand the relationship between numbers, even when those numbers are unknown.

Variables in Algebra

Variables in algebra are like blank spaces we fill with numbers through substitution. They are symbols, usually letters like \(x\), \(y\), or \(t\), that represent unknown or changeable numbers. In algebra, variables help us generalize mathematical patterns and express rules or relationships in a flexible way. For our example, the variable is \(t\).

Here's how variables function in expressions:

• They allow expressions to represent multiple problems and solutions at once.
• They can be changed with different numbers to find various results.
• Variables make it easier to communicate complex problems in a simple form.

Variables are essential in algebra because they enable us to work with unknowns, allowing us to derive solutions through processes like substitution, as we did in the original exercise.