Problem 6 - Precalculus: Mathematics for Calculus

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.

Step 1: Identify the Number of Equations and Variables

The given system of equations consists of three equations with three variables: \(x\), \(y\), and \(z\).

Step 2: Check for Linear Structure in Each Equation

Examine each equation to see if it fits the form \(a_1x + a_2y + a_3z = d\), where \(a_1, a_2, a_3\), and \(d\) are constants, and \(x, y, z\) are variables.1. \(x - 2y + 3z = 10\), which can be rearranged as \(1x + (-2)y + 3z = 10\).2. \(2x + 5y = 2\), already in linear form. The coefficient of \(z\) is zero.3. \(y + 2z = 4\), which can be written as \(0x + 1y + 2z = 4\).

Step 3: Confirm Linearity by Analyzing Equations

Linear equations do not have variables multiplied together, exponents other than one, or contain any other form of non-linear terms. All given equations fit the criteria.Equation 1: Variables \(x\), \(y\), and \(z\) are of first degree and not multiplied together.Equation 2: Same with variables \(x\) and \(y\).Equation 3: Same with variables \(y\) and \(z\).

Step 4: Final Verification

Since all three equations are of the form \(a_1x + a_2y + a_3z = d\), we can confirm that each equation individually, and the whole system collectively, is linear.

Key Concepts

Systems of Equations

A system of equations refers to a collection of two or more equations that contain the same set of variables. The goal is typically to find the values of these variables that satisfy all the equations in the system simultaneously. In our original exercise, we are dealing with a system of three equations involving three variables: \(x\), \(y\), and \(z\).

• The first equation is \(x - 2y + 3z = 10\).
• The second equation is \(2x + 5y = 2\).
• The third equation is \(y + 2z = 4\).

All these equations are linear, meaning there are no squared terms, no variables multiplied by each other, and each variable appears only to the first power. Solving a system of equations helps us find the exact values for all the variables that make every equation in the system true.
Understanding systems of equations is crucial for solving problems in fields like physics, engineering, and economics, where multiple conditions must be satisfied concurrently.

Variables

Variables are symbols that represent unknown values and are the building blocks of equations, allowing us to formulate and solve problems mathematically. In our system of equations, \(x\), \(y\), and \(z\) are the variables. Each one plays a unique role in the equations as placeholders for potentially different numbers.

• \(x\), \(y\), and \(z\) are independent, meaning the value of one does not directly determine the values of the others without solving the system.
• Variables can often represent quantities like distance or time in practical applications, which allows the mathematical model to mimic real-world scenarios.

Recognizing that variables can carry different values under different conditions helps in solving the system accurately. It is essential to determine the correct values of our variables through methods like substitution or elimination to unlock the solution for the entire system of equations.

Coefficients

Coefficients are the numerical or constant multipliers of the variables in an equation. They play a vital role in defining the relationship between different variables in a linear equation. In our system of equations:

• The coefficient of \(x\) in the first equation is 1.
• In the second equation, the coefficient of \(x\) is 2.
• The coefficients for \(y\) are -2, 5, and 1 across the three equations.
• The coefficients for \(z\) are 3, 0, and 2, as presented in respective equations.

Each coefficient affects how much weight or influence each variable has within its equation. For example, a larger coefficient means the variable associated with it has a more substantial impact in achieving equality in that equation. Understanding the role of coefficients is crucial for manipulating and solving linear systems, such as employing methods in algebra to substitute or eliminate variables effectively.

Linearity

Linearity in equations implies that all terms are either constants or constants multiplied by a single variable. Linear equations are represented in the general form \(a_1x + a_2y + a_3z = d\), where \(a_1, a_2, a_3\), and \(d\) are constants, and each equation does not have any variable raised to a power other than one.
In the original problem, each equation exhibits this linear form by having each term as a product of a constant and a variable or a constant alone. There are no terms involving squares, cubes, or products of different variables. This ensures that the relationships defined by the equations are predictable and result in straight lines or flat planes when graphed.
The property of linearity allows for several straightforward methods to solve these systems, such as graphing, substitution, or elimination. Linearity makes the equations much easier to handle analytically as well and is a fundamental feature that helps predict and analyze behaviors in various scientific and engineering contexts.