Problem 5 - Algebra and Trigonometry

Question

\(3-6\) . State whether the equation or system of equations is linear. $$ \left\\{\begin{aligned} x y-3 y+z &=5 \\ x-y^{2}+5 z &=0 \\ 2 x &+y z=3 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system of equations is nonlinear due to terms like \(xy\), \(y^2\), and \(yz\).

Step 1: Review the System of Equations

The given system of equations consists of three equations:1. \( xy - 3y + z = 5 \)2. \( x - y^2 + 5z = 0 \)3. \( 2x + yz = 3 \).

Step 2: Identify Nonlinear Components

To determine whether the system is linear, examine each equation for nonlinear terms.- In Equation 1: The term \( xy \) is a product of variables, making it nonlinear.- In Equation 2: The term \( y^2 \) involves a squared variable, making it nonlinear.- In Equation 3: The term \( yz \) is a product of variables, making it nonlinear.

Step 3: Conclude Linearity

Since each equation in the system contains nonlinear terms (either products of variables or squared variables), the entire system is nonlinear. A system of equations is linear only if all equations are linear, i.e., no products, powers, or other nonlinear operations on the variables.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a shared set of variables. When solving such systems, the goal is typically to find the values of the variables that satisfy all the equations simultaneously.

In our example, we are dealing with three equations:

• Equation 1: \( xy - 3y + z = 5 \)
• Equation 2: \( x - y^2 + 5z = 0 \)
• Equation 3: \( 2x + yz = 3 \)

Each equation involves the variables \( x, y, \) and \( z \). To understand and solve the system, careful analysis of each equation considering its variables and operations is essential. Identifying the nature of the operations (linear or nonlinear) helps in determining the best approach for solving the system.

Nonlinear Equations

Nonlinear equations are equations in which the variables do not only appear to the first power and are not involved in simple addition or subtraction.

In a linear context, each term in the equation is a constant or product of a constant and a first-degree variable. However, the presence of products of variables like \( xy \) or powers such as \( y^2 \) makes an equation nonlinear.

In our system:

• Equation 1 has the term \( xy \), which is a product of two variables.
• Equation 2 includes \( y^2 \), indicating a squared variable.
• Equation 3 has the term \( yz \), another product of variables.

These characteristics render the entire system nonlinear, making it clear that it cannot be treated using methods suitable for linear equations, such as matrix methods.

Variables Multiplication

Variables multiplication happens when two or more different variables are multiplied together, as seen in terms like \( xy \) and \( yz \).

Such terms are fundamentally nonlinear because they do not represent a single linear transformation or addition/subtraction of variables. They introduce complexity due to the way different variables are interdependent in these products.

Terms like \( xy \) in Equation 1 and \( yz \) in Equation 3 demonstrate how different variables can interact. This interaction often makes solving the equation analytically more challenging, compared to equations with purely linear terms.

Squared Variables

Squared variables, such as \( y^2 \), involve raising a variable to the power of two. This action significantly impacts the linearity of an equation.

The term \( y^2 \) seen in Equation 2 signifies that the variable \( y \) affects the equation's outcome in a nonlinear manner, as its value impacts the equation more dramatically than if it were just \( y \).

While linear equations allow each variable to affect the result independently in a predictable manner, squared variables create exponential dependency. As a result, the equation involves complexities in behavior and solution interpretation, characteristic of nonlinear equations.