Consider the expressions 3(T-1) and 3 T-3. a. Copy the...

Consider the expressions $3(T-1)$ and $3 T-3$. a. Copy the table, and complete the first four columns. In the last four columns, choose your own input values and calculate the output for both rules. Use fractions for at least two of your input values. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \boldsymbol{T} & 4 & & 100 & & & & & \\ \hline \mathbf{3} \boldsymbol{T}-\mathbf{3} & & 30 & & 87 & & & & \\ \hline \mathbf{3}(\boldsymbol{T}-\mathbf{1}) & 9 & 30 & & & & & & \\ \hline \end{array} $$ b. Show that $3(T-1)$ and $3 T-3$ are equivalent expressions. c. Find two more expressions that are equivalent to $3(T-1)$ and $3 T-3$.

Consider the expressions $3(T-1)$ and $3 T-3$.
a. Copy the table, and complete the first four columns. In the last four columns, choose your own input values and calculate the output for both rules. Use fractions for at least two of your input values.
$$
\begin{array}{|l|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{T} & 4 & & 100 & & & & & \\
\hline \mathbf{3} \boldsymbol{T}-\mathbf{3} & & 30 & & 87 & & & & \\
\hline \mathbf{3}(\boldsymbol{T}-\mathbf{1}) & 9 & 30 & & & & & & \\
\hline
\end{array}
$$
b. Show that $3(T-1)$ and $3 T-3$ are equivalent expressions.
c. Find two more expressions that are equivalent to $3(T-1)$ and $3 T-3$.

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Key Concepts

Function Evaluation Using Tables

Using tables to evaluate functions involves choosing specific input values for the variable and computing the corresponding outputs. This method is useful for comparing how different rules or expressions behave across various inputs and for verifying that the expressions in question are equivalent.

Algebraic Manipulation

Algebraic manipulation involves techniques such as expansion, simplification, and rearrangement of terms to transform and compare expressions. This process is key in identifying different forms of the same underlying relationship, as seen when finding additional expressions equivalent to 3(T - 1) and 3T - 3.

Distributive Property

The distributive property allows multiplication to be distributed over addition or subtraction within parentheses, meaning that a(b + c) is equivalent to ab + ac. This concept is used to show that expressions like 3(T - 1) can be rewritten as 3T - 3 by multiplying 3 with each term inside the parentheses.

Equivalent Expressions

Equivalent expressions are different algebraic expressions that yield the same value for every possible value of their variables. Demonstrating that 3(T - 1) and 3T - 3 are equivalent involves using algebraic techniques like the distributive property, which confirms that both expressions simplify to the same form.

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