Problem 9 - Algebra Form and Function

Question

For each of the linear expressions in \(x\) in exercises. give the constant term and the coefficient of \(x\). $$ 3 x+4 $$

Step-by-Step Solution

Answer

Answer: In the linear expression 3x + 4, the constant term is 4 and the coefficient of x is 3.

Step 1: Identify the constant term

In the expression \(3x+4\), the constant term is the term that does not have a variable (i.e., \(x\)), which is 4 in this case.

Step 2: Identify the coefficient of \(x\)

In the expression \(3x+4\), the term containing \(x\) is \(3x\). The number multiplying the variable \(x\) is called the coefficient, and in this case, the coefficient of \(x\) is 3.

Step 3: Present the results

In the linear expression \(3x+4\), the constant term is 4 and the coefficient of \(x\) is 3.

Key Concepts

constant term

In a linear expression, the constant term is the number that stands alone without any variable attached to it. It is important to recognize because it represents a fixed value within the expression. For example, in the linear expression \(3x + 4\), the constant term is the number 4. Here, 4 is not influenced by the variable \(x\); it remains unaffected regardless of the value \(x\) takes. This term can be thought of as a stand-alone number that you would always add to or subtract from the expression, once the variable part is evaluated.

• Easily identifiable as it lacks a variable.
• Represents a static value no matter what.
• Adds more depth to the expression beyond the variable interaction.

coefficient

The coefficient in a linear expression is strictly related to the variable. It is the numerical factor that you multiply with the variable. Being able to identify the coefficient is crucial because it tells you how much the value of the expression will change with each unit change of the variable. Taking the example \(3x + 4\), here the coefficient of \(x\) is 3. This suggests that for every single unit increase in \(x\), the term \(3x\) will increase by 3 units.

• Always found directly attached to the variable.
• Determines the rate of change for the expression.
• Helps in understanding the strength of the variable’s impact within the equation.

variables

Variables are symbols in expressions and equations that stand in for unknown values or numbers. They are like placeholders or figures that can change. Recognizing variables is vital as they are the part of the expression that will have varying impacts on the outcome based on their values. In the expression \(3x + 4\), \(x\) is a variable. It means the value of the expression can change depending on what number is plugged in for \(x\).

• Serve as a foundation for formulating equations and expressions.
• Represent values that can shift, altering the expression’s result.
• Used extensively in algebra to express very adaptable and changing solutions.