Problem 94 - Intermediate Algebra

Question

Complete the table. $$ T=x-1.5 $$ $$ \begin{array}{|c|c|} \hline x & T \\ \hline 3.7 & \\ \hline 10 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

For \( x = 3.7 \), \( T = 2.2 \) and for \( x = 10 \), \( T = 8.5 \).

Step 1: Understand the Formula

The formula given is \( T = x - 1.5 \). Here, \( T \) is expressed in terms of \( x \). This means you subtract 1.5 from any value of \( x \) to find the corresponding value of \( T \).

Step 2: Calculate for \( x = 3.7 \)

Substitute \( x = 3.7 \) into the formula: \( T = 3.7 - 1.5 \). Calculate the result: \( T = 2.2 \). Thus, the value of \( T \) corresponding to \( x = 3.7 \) is 2.2.

Step 3: Calculate for \( x = 10 \)

Substitute \( x = 10 \) into the formula: \( T = 10 - 1.5 \). Calculate the result: \( T = 8.5 \). Thus, the value of \( T \) corresponding to \( x = 10 \) is 8.5.

Key Concepts

Function Evaluation

To evaluate a function is to determine the output when given an input. In our exercise, the function is given by the formula \( T = x - 1.5 \). Here, \( T \) is the output or the value of the function, while \( x \) is the input. The process of function evaluation involves the following steps:

• Plugging in the value of the input, \( x \), into the function. In our case, you replace \( x \) with the specific number that you have in mind.
• Perform the operations specified in the function with this number. For example, subtract 1.5 from the input value as given by our specific equation.
• Calculate the result to find the output of the function, which is \( T \) in this case.

Function evaluation is a fundamental concept in mathematics that shows how functions work to transform inputs into outputs.

Substitution Method

The substitution method is crucial for solving equations like the one in our example. In simple terms, substitution means replacing a variable with a number or another expression. Let’s see how it is used here:To find the value of \( T \) when \( x = 3.7 \), substitute 3.7 into the equation \( T = x - 1.5 \). This gives us:\(T = 3.7 - 1.5\)Similarly, for \( x = 10 \), substitute 10 in place of \( x \):\(T = 10 - 1.5\)The substitution method simplifies the process of finding unknowns by making it clear which value needs to be used in the equation. It’s especially handy when working with multiple variables and equations, allowing for straightforward solutions through replacing and solving.

Arithmetic Operations

Arithmetic operations comprise the basic calculations such as addition, subtraction, multiplication, and division. In the context of our problem, subtraction is the key operation. Here’s how subtraction applies:- For \( x = 3.7 \), we perform the subtraction by calculating \( 3.7 - 1.5 \), resulting in 2.2.- For \( x = 10 \), the subtraction is \( 10 - 1.5 \), which equals 8.5.These simple operations are fundamental to evaluating functions and solving equations. It’s essential to keep accuracy in calculations to ensure the right result. Through arithmetic, we can manipulate numbers to find required solutions, demonstrating the practicality and necessity of these operations in mathematics.