Problem 3 - Algebra 1: Concepts and Skills

Question

Tell whether each equation is linear or not linear. Explain your answer. $$a^{2}+1=9$$

Step-by-Step Solution

Answer

The equation \(a^{2}+1=9\) is not linear because its variable 'a' is of second degree, not the first degree.

Step 1: Identify the degree of the variable in the equation

The first step is to identify the degree of the given equation. In our equation \(a^{2}+1=9\), the degree of the variable 'a' is 2.

Step 2: Compare the degree of the equation with the condition of linearity

A linear equation must be of first degree i.e., the highest exponent of the variable should be 1. In our equation, the highest exponent of 'a' is 2, which is more than 1.

Step 3: Conclude if the equation is linear or not

Since the degree of the given equation is not 1, we can conclude that the equation \(a^{2}+1=9\) is not linear.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a crucial concept that indicates the highest power of the variable in the expression. For any given polynomial, the degree tells you the most times the variable is multiplied by itself. For example, in the expression \( a^2 + 1 = 9 \), the highest exponent of the variable 'a' is 2, so it is called a second-degree polynomial. This is important because the degree helps determine the nature of the equation—specifically whether it might be linear or not.

• Degree is determined by the largest exponent on the variable.
• A polynomial of degree 0 is a constant.
• Degree 1 polynomials are linear.

Understanding the degree will give you insights into the possible shape of the graph of the equation and will influence the methods used for solving it.

Linearity Condition

When discussing equations, they are classified based on their degree and the behavior of their graph. The linearity condition specifically refers to whether the degree of the polynomial is exactly 1. A linear equation takes the form \( ax + b = 0 \), where \( a \) and \( b \) are constants, and the variable x is raised to the power of 1.

• A linear equation's graph is always a straight line.
• Linearity ensures the relationship between variables is proportional.
• If any term has a degree higher than 1, the equation is not linear.

Thus, an equation like \( a^2 + 1 = 9 \) does not satisfy the linearity condition since the degree of 'a' is not 1 but 2. This makes it a quadratic equation instead of a linear one.

Identifying Linear vs Non-linear Equations

Identifying whether an equation is linear or non-linear is a fundamental skill in algebra. This involves checking the highest degree of the variable. If the degree is 1, it's linear; if it's more than 1 or involves operators like trigonometric functions, it's non-linear.

• Linear equations have variables raised only to the first power.
• Non-linear equations include quadratics \( (x^2) \), cubics \( (x^3) \), etc.
• Graphically, linear equations form straight lines; non-linear ones form curves.

For example, in the equation \( a^2 + 1 = 9 \), the presence of \( a^2 \) confirms it's non-linear. Recognizing the type of equation helps in choosing the appropriate solution methods and predicting the graph's shape.