Problem 3 - Algebra and Trigonometry

Question

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 is not linear.

Step 1: Identify the Type of Equations

Examine each equation in the system to determine if it is linear. A linear equation can be written in the form \( ax + by + cz + ext{(constant)} = 0 \), where \(a\), \(b\), and \(c\) are constants, and the variables \(x\), \(y\), and \(z\) are not raised to any powers other than one.

Step 2: Analyze the First Equation

Consider the first equation: \(xy - 3y + z = 5\). This equation has a term \(xy\), which is a product of two variables. Since linear equations cannot have products or powers of variables, this equation is not linear.

Step 3: Examine the Second Equation

Look at the second equation: \(x - y^2 + 5z = 0\). Here, \(y^2\) is the square of a variable, which violates the requirement for linear equations to have variables only to the first power. Thus, this equation is not linear.

Step 4: Evaluate the Third Equation

Evaluate the third equation: \(2x + yz = 3\). It contains the term \(yz\), which is a product of two variables. As linear equations cannot involve multiple variables multiplied together, this equation is not linear.

Step 5: Conclude the System's Linearity

Determine whether the overall system is linear by checking each constituent equation. Since none of the equations is linear, the system of equations is not linear.

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations, each with one or more variables. The goal is to find values for the variables that satisfy all the equations simultaneously. Often, these systems arise in real-world problems where multiple relationships need to be maintained.
For instance:

• Two linear equations with two variables can be visualized as straight lines on a graph.
• The solution is the point(s) where these lines intersect.

Systems of equations can be classified based on their nature into linear and non-linear systems, as well as those which have no solution, one solution, or infinitely many solutions. Understanding the types of systems helps to determine the method of solving them, whether through substitution, elimination, or graphical methods.

Non-Linear Equations

Non-linear equations are equations in which the variables are not simply multiplied by constants and added together. In other words, the variables may be raised to a power other than one or may involve products of variables.
Common characteristics of non-linear equations include:

• Terms with squared variables, such as \(y^2\), cubic terms, exponential terms, or variables multiplied together, like \(xy\).
• These equations often represent curves rather than straight lines when graphed.

In the given exercise, each equation contains terms like \(xy\), \(y^2\), and \(yz\), indicating their non-linear nature. Solutions to non-linear systems can be more complex and may require numerical methods or special solution techniques like factoring or graphing.

Variables and Constants

Variables and constants are fundamental components of algebraic equations. Understanding their roles is key to solving equations effectively.

• Variables are symbols used to represent numbers whose values are not yet determined. Commonly denoted by letters such as \(x\), \(y\), and \(z\), variables can vary and take on different values depending on the situation or the system of equations in which they are used.
• Constants are fixed values that do not change and are usually denoted by numbers such as 3, 5, or -2. They represent known quantities within the equation.

In equations, constants adjust the balance or the relationship represented by the variables. For instance, in the term \(3y\) from the exercise, 3 is a constant, and \(y\) is the variable that can change, determining the variable's role in solving the equation.