Problem 69 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ -3(t)(t) \text { when } t=-1 $$

Step-by-Step Solution

Answer

The value of the expression \(-3(t)(t)\) when \(t = -1\) is \(-3\).

Step 1: Understanding the Expression

In the expression \(-3(t)(t)\), there are two terms which are the same (which is \(t\)) and they are being multiplied together. This expression can be rewritten as \(-3t^2\). It's important to note that \(t\) is a variable here and it can be replaced by any value which is given.

Step 2: Substituting Value of Variable

We are given that \(t = -1\). So, our task now is to substitute the value of \(t\) into the expression that was previously simplified into \(-3t^2\). This gives us \(-3(-1)^2\).

Step 3: Evaluating the Expression

Now, we simply have to calculate the value for \(-3(-1)^2\). According to order of operations, we first have to square the \(-1\) to get \(1\) and then multiply it by \(-3\). This gives us \(-3 \times 1\) which equals \(-3\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra. It involves replacing a variable in an expression with a given number. In our exercise, we are asked to evaluate an expression where the variable is given a specific value. Imagine the variable as a placeholder; once you substitute the number for the variable, it allows you to perform numerical calculations.

In the expression \(-3(t)(t)\), the variable \(t\) is present twice, meant to be multiplied by itself. Given that \(t=-1\), you replace both instances of \(t\) in the expression with \(-1\). This leads to a new numerical expression: \(-3(-1)(-1)\).

Substitution is the first step to solving the problem because it converts an abstract expression into something that can be computed numerically. Doing this correctly paves the way for further steps like applying the order of operations.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which different operations should be performed. A handy acronym to remember is PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and square roots, etc.)
• M: Multiplication and D: Division (left to right)
• A: Addition and S: Subtraction (left to right)

In our expression \(-3(-1)^2\), after substitution, we follow PEMDAS to solve it correctly.

First, you handle exponents before multiplication. We see the exponent is \((-1)^2\). By squaring \(-1\), you get \(1\). After calculating the exponent, you then multiply \(-3\) by \(1\), which simplifies to \(-3\). This order must be respected to ensure accurate results.

Exponents

Exponents are shorthand for repeated multiplication of the same factor. In our example, \((-1)^2\) represents \(-1\) multiplied by \(-1\), resulting in \(1\).

Understanding how exponents work is crucial when evaluating expressions. A positive exponent indicates how many times a base number is multiplied by itself.

• For example, in the expression \((-1)^2\), \(-1\) is the base and \(2\) is the exponent.
• This tells us to multiply \(-1\) by itself \(2\) times.

The computation, \((-1) imes (-1)\), results in \(1\), because multiplying two negative numbers results in a positive number.

Without correctly applying the rules for exponents, the outcome of evaluating expressions with variables could be incorrect. Being clear on how exponents transform an expression helps simplify the problem efficiently.