Problem 4 - Algebra and Trigonometry

Question

State whether the equation or system of equations is linear. $$ \left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 y &=2 \\ y+2 z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system of equations is linear because each equation is linear.

Step 1: Understand Linear Equations

In a linear equation, each term is either a constant or the product of a constant and a single variable. Linear equations do not have variables raised to any power other than one.

Step 2: Analyze Each Equation

Inspect each equation in the system given. The first equation is \( x - 2y + 3z = 10 \). This is linear because all variables are first-degree (they have no exponents other than 1).The second equation is \( 2x + 5y = 2 \). This is also linear for the same reason.The third equation is \( y + 2z = 4 \). Again, this equation is linear.

Step 3: Verify the System

Since each individual equation is linear, the entire system of equations is linear. There are no variables multiplied by each other and no variables raised to power(s) other than 1.

Key Concepts

Understanding Systems of Equations

A system of equations consists of multiple equations working together. These equations involve the same set of variables. In math, systems of equations are used to find values for each variable that satisfy all equations simultaneously.
A simple example is when you have two equations like:

• \( x + y = 6 \)
• \( x - y = 2 \)

The goal is to find the values of \( x \) and \( y \) that make both of these statements true at the same time. To solve a system, you can use methods like substitution, elimination, or graphing.
In the context of linear equations, as in the original exercise, a system of equations is called a 'linear system' if all involved equations are linear. Linear systems are quite common in real-world applications, including engineering and economics.

The Role of Variables in Equations

Variables are symbols that represent unknown values in equations. They act like placeholders that can be filled with different numbers. You've likely seen variables represented as letters like \( x \), \( y \), or \( z \).
In the system of equations given, there are variables \( x \), \( y \), and \( z \). Each variable can represent a different unknown value, which we need to solve for. This is where the purpose of the system of equations comes into play to determine these values.
Understanding variables enables solving equations effectively. In linear equations, any variables included are raised only to the first power, maintaining simplicity and making these equations straightforward to solve.

Exploring the Degree of Polynomial

The degree of a polynomial is an important concept that helps to define its nature. It's determined by the highest power of the variable within an expression. In linear equations, this highest power is always 1, which is why they are easier to work with.
For example, the first equation in the system, \( x - 2y + 3z = 10 \), is linear because the variables \( x \), \( y \), and \( z \) each have a degree of 1. There are no variables squared or cubed, which makes it manageable to solve.
Understanding the degree of a polynomial is crucial as it dictates the methods you'll use for solving the equation. Linear equations, with a degree of one, often allow for straightforward solution techniques and are foundational in studying more complex algebraic topics.