Problem 4 - Algebra and Trigonometry

Question

\(3-6\) . State whether the equation or system of equations is linear. $$ x^{2}+y^{2}+z^{2}=4 $$

Step-by-Step Solution

Answer

The equation is not linear.

Step 1: Identify the Equation

Look at the given equation: \[ x^2 + y^2 + z^2 = 4 \]Identify the highest power for each of the variables \(x\), \(y\), and \(z\). This will determine whether it is linear or not.

Step 2: Analyze the Highest Power

The highest power of any variable in the equation is 2 because \(x^2\), \(y^2\), and \(z^2\) each appear in quadratic form. In a linear equation, the highest power should be 1 for each variable.

Step 3: Determine Linearity

Since the highest power of the variables is 2, which is greater than 1, the equation is not linear. A linear equation should not have any variables raised to a power other than 1.

Key Concepts

quadratic equations

Quadratic equations are equations in which the highest power of the variable is 2. This means they contain terms like \( x^2 \), \( y^2 \), or \( z^2 \). Unlike linear equations, where the graph represents a straight line, quadratic equations graph as a curve called a parabola. These parabolas can open upward or downward, depending on the sign in front of the quadratic term.

In a quadratic equation, the general form is usually written as:

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

The presence of the \( x^2 \) term is what distinguishes these equations from linear ones. Solving a quadratic equation often involves finding the values of \( x \) when the equation equals zero, also known as finding the roots. Methods to solve quadratic equations include factoring, using the quadratic formula, or completing the square.

powers of variables

The concept of powers of variables refers to the exponents associated with variable terms in an equation. When we analyze equations, identifying the power of each variable is crucial because it helps us understand the equation's nature, such as whether it's linear or quadratic.

For example, in the equation \( x^2 + y^2 + z^2 = 4 \), each term has a power of 2, indicating a quadratic form. If each variable had a power of 1, like in \( x + y + z = 4 \), it would be linear.

• Linear terms: Variables with the highest power of 1, e.g., \( x \), \( y \), or \( z \).
• Quadratic terms: Variables with a power of 2, e.g., \( x^2 \), \( y^2 \), or \( z^2 \).

Understanding the power of a variable is significant because it determines the degree of the equation, influencing the graph's shape and the solution methods used.

nonlinear equations

Nonlinear equations are equations that do not form a straight line when graphed. In these types of equations, at least one variable has an exponent greater than 1. Because of this, the relationships they describe are not constant and change depending on the specific values of the variables involved.

An example is the equation \( x^2 + y^2 + z^2 = 4 \). Since each variable is squared, the equation is nonlinear. Nonlinear equations cannot be represented as a simple line but often manifest as curves or other shapes on a graph.

• Cubic equations: Equations where the highest degree is 3, e.g., \( x^3 \).
• Quadratic equations: Highest degree is 2, e.g., \( x^2 \).
• Others: May have degrees 4 (quartic), 5 (quintic), etc.

Solving nonlinear equations can be more complex than solving linear equations due to the variety of shapes and interactions involved. Various techniques are used, including graphing, substitution, and numerical methods, to find solutions to these equations.