Problem 7 - Algebra Form and Function

Question

Is the expression linear? $$ 4^{2}+(1 / 3) x $$

Step-by-Step Solution

Answer

Explain. Answer: Yes, the expression is linear because the power of the variable 'x' is 1 and it contains a constant term $$4^2$$. The expression fulfills the properties of a linear expression in the form of $$ax + b$$.

Step 1: Identify the components of the expression

In the given expression, $$4^2 + \frac{1}{3}x$$, we can identify two components: - $$4^2$$, which is a constant term; and - $$(\frac{1}{3})x$$, which is a term with a variable 'x' in it. Now let's analyze these components to check if they fulfill the properties of a linear expression.

Step 2: Check for the variable's power

In the term with the variable, $$(\frac{1}{3})x$$, the power of the variable 'x' is 1, which meets the condition for a linear expression.

Step 3: Check for the constant term

In the constant term, $$4^2$$, it represents a constant value since it does not contain any variable. Therefore, it meets the condition for a linear expression as well. Since both the components in the expression meet the conditions of a linear expression, we can conclude that the given expression is a linear expression.

Key Concepts

Constant Term

In algebraic expressions, a constant term is a number on its own. It doesn't include any variables. In our given expression, \(4^2\) acts as the constant term. To identify a constant term, look for numerical values that don't change. For instance:

• In \(4x + 7\), the constant term is 7 because it isn't multiplied by any variable.
• In \(5 + 9y\), 5 is the constant term.

The value remains the same no matter what the variable represents. This is what makes it 'constant'. Constant terms are important in forming linear expressions because they don't change the expression's degree. They simply add or subtract from the output without affecting the variable's behavior.
Recognizing constant terms helps in simplifying expressions and solving equations. It makes calculations straightforward and ensures that solutions can be found by focusing on the changing variables.

Variable Power

When we talk about linear expressions, the power of the variable is crucial. The variable's power determines the nature of the expression. In our expression \((\frac{1}{3})x\), the variable 'x' has a power of 1. A linear expression is defined by having a variable with this power. Here's why:

• Linear expressions are the simplest polynomials, usually in the form \(ax + b\).
• They graph as straight lines on a coordinate plane.
• Linear equations change at a constant rate.

Higher powers, like squared (\(x^2\)), create curves and aren't linear. Keeping the variable's power at 1 keeps the expression straightforward and predictable. When solving problems, always check the variable's power to determine the type of expression. Understanding this helps you know how the equation or expression will behave, making it easier to analyze and solve.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication). The expression \(4^2 + \frac{1}{3}x\) is an example. In an algebraic expression:

• Variables represent unknown values. They allow the expression to model real-world situations.
• Constants are fixed values that give the expression its specific nature.
• Operations describe how the terms are combined, such as with addition or subtraction.

An understanding of algebraic expressions will help you identify different types, like linear expressions. An expression is 'linear' if its highest variable power is 1. Recognizing algebraic expressions helps in formulating equations and functions that solve mathematical problems.
Mastering these expressions is a significant step in learning algebra, enabling you to describe and understand complex mathematical relationships. Practicing with various expressions allows you to refine your problem-solving skills and think more abstractly.