Problem 1 - Intermediate Algebra

Question

Fill in the blanks. \(\left\\{\begin{array}{l}2 x+y-3 z=0 \\ 3 x-y+4 z=5 \\ 4 x+2 y-6 z=0\end{array}\right.\) is called a _____ of three linear equations in three variables. Each equation is written in ____ \(A x+B y+C z=D\) form.

Step-by-Step Solution

Answer

The system is called a "system of three linear equations in three variables" and each equation is in "standard" form.

Step 1: Understand the System of Equations

We are given a set of three equations, each involving three variables: \(x\), \(y\), and \(z\). Together, they form a system of equations that we need to identify. Systems like these are typically classified by the number of equations and variables present.

Step 2: Identify the System Type

The given set consists of three equations with three variables, \(x\), \(y\), and \(z\). Such a system is commonly referred to as a "system of three linear equations in three variables." This type of system can be solved by finding values of \(x\), \(y\), and \(z\) that satisfy all three equations.

Step 3: Understand the Standard Form of Each Equation

Each equation in the system is presented in a specific format, \(A x + B y + C z = D\). This standard form for linear equations signifies that each equation is a linear combination of the three variables set equal to a constant. All equations in this set adhere to this format.

Step 4: Fill in the Blanks

From the understanding in steps 1 through 3, the system is called a "system of three linear equations in three variables" and each equation is written in the "standard" \(A x + B y + C z = D\) form.

Key Concepts

Linear Equations

Linear equations form the basis of systems of linear equations, which are fundamental in algebra and numerous applications.
A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. It is characterized by the presence of one or more variables, each raised to the power of one. Here are some key points about linear equations:

• Linear implies that the equation graphs to a straight line, meaning there are no curves or turns involved.
• Each term in a linear equation involves a variable, with no exponents greater than one.
• In the context of systems, linear equations can be combined to form equations with multiple variables.

Understanding linear equations helps in recognizing how they can represent relationships within systems and how solutions to these equations reflect real-world situations.

Variables

Variables play a crucial role in linear equations and systems of linear equations. They are the unknowns that we aim to solve for.A variable is represented by a symbol, commonly letters like \(x\), \(y\), and \(z\) in equations:

• Variables act as placeholders for values that need to be determined to solve equations.
• In a system of equations, different equations might involve the same variables. This allows the equations to be solved simultaneously to find a consistent solution for all included variables.
• Understanding variables is essential for solving not only linear equations but also more complex mathematical concepts.

By manipulating linear equations and interpreting the roles of variables, students can uncover values that satisfy all parts of a given system.

Standard Form

The standard form of a linear equation is a significant format that simplifies both solving and analyzing these mathematical expressions.
In the context of systems, the standard form is pivotal.In linear algebra, a linear equation in standard form appears as \(A x + B y + C z = D\):

• This notation helps to uniformly arrange linear equations for easier comparison and manipulation.
• Terms \(A\), \(B\), \(C\), and \(D\) are constants, where \(A\), \(B\), and \(C\) multiply the variables.
• A standard form represents a balance between the variables on one side of the equation and a constant on the other, creating an equation ready for various solving methods, such as substitution or elimination.

Adopting the standard form facilitates understanding how equations relate to each other and provides a framework for more advanced exploration in mathematics.