Problem 29 - Algebra A Combined Function

Question

Solve. See the Concept Check in this section. For the system \(\left\\{\begin{aligned} x \quad+z &=7 \\ y+2 z &=-6, \text { which is the correct corresponding matrix? } \\ 3 x-y \quad &=0 \end{aligned}\right.\) a. \(\left[\begin{array}{rrr}1 & 1 & 7 \\ 1 & 2 & -6 \\ 3 & -1 & 0\end{array}\right]\) b. \(\left[\begin{array}{rrrr}1 & 0 & 1 & 7 \\ 1 & 2 & 0 & -6 \\ 3 & -1 & 0 & 0\end{array}\right]\) C. \(\left[\begin{array}{rrrr}1 & 0 & 1 & 7 \\ 0 & 1 & 2 & -6 \\ 3 & -1 & 0 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

The correct corresponding matrix is option (c).

Step 1: Understand the System of Equations

First, identify the variables in each equation of the given system:1. Equation 1: \( x + z = 7 \)2. Equation 2: \( y + 2z = -6 \)3. Equation 3: \( 3x - y = 0 \)Here, we have three variables: \( x \), \( y \), and \( z \). Each equation may or may not have certain variables missing, which will affect how the matrix is constructed.

Step 2: Identify Matrix Structure

A corresponding matrix for a system of linear equations consists of the coefficients of the variables in the system. It is helpful to organize the matrix with each row corresponding to one equation. The columns represent the coefficients of \( x \), \( y \), and \( z \), followed by the constants on the right side of the equation conversion as the last column.

Step 3: Create an Augmented Matrix

Write out the augmented matrix from the system:1. From \( x + z = 7 \), note: coefficient of \( x \) is 1, \( y \) is 0 (since it's missing), \( z \) is 1. Thus, the row is \([1, 0, 1, 7]\).2. From \( y + 2z = -6 \), note: \( x \) is 0, \( y \) is 1, \( z \) is 2. Thus, the row is \([0, 1, 2, -6]\).3. From \( 3x - y = 0 \), note: \( x \) is 3, \( y \) is -1, \( z \) is 0. Thus, the row is \([3, -1, 0, 0]\).

Step 4: Compare with Given Options

Compare the matrix you formed, \(\begin{bmatrix}1 & 0 & 1 & 7 \0 & 1 & 2 & -6 \3 & -1 & 0 & 0 \\end{bmatrix} \), against the options provided:- a. \(\begin{bmatrix}1 & 1 & 7 \1 & 2 & -6 \3 & -1 & 0 \\end{bmatrix}\)- b. \(\begin{bmatrix}1 & 0 & 1 & 7 \1 & 2 & 0 & -6 \3 & -1 & 0 & 0 \\end{bmatrix}\)- c. \(\begin{bmatrix}1 & 0 & 1 & 7 \0 & 1 & 2 & -6 \3 & -1 & 0 & 0 \\end{bmatrix}\)The correct matrix matches option (c).

Step 5: Conclusion: Correct Matrix Choice

The correct corresponding matrix for the system of equations is option (c): \[\begin{bmatrix}1 & 0 & 1 & 7 \0 & 1 & 2 & -6 \3 & -1 & 0 & 0 \\end{bmatrix}\]

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In this particular case, the system consists of three equations: \( x + z = 7 \), \( y + 2z = -6 \), and \( 3x - y = 0 \). Each equation represents a linear relationship between the variables \( x \), \( y \), and \( z \).

Solving a system of equations involves finding the values for the variables that satisfy all the equations simultaneously. These solutions can often represent intersections of lines or planes in a space.

Common methods for solving systems of equations include substitution, elimination, and using matrices. Substitution involves expressing one variable in terms of others and substituting it into the remaining equations. Elimination focuses on canceling out a variable by adding or subtracting equations. Using matrices involves transforming the equations into a matrix form and then simplifying to find the solution.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors and operations on vectors, matrices, and linear transformations. It is foundational in understanding systems of linear equations, as it provides tools and methods to analyze multi-variable equations efficiently.

Key concepts in linear algebra include:

• Vectors: Objects that have magnitude and direction, often represented as arrays of numbers.
• Matrices: Rectangular arrays of numbers or functions that can represent systems of linear equations.
• Determinants: Values that can indicate whether a matrix has an inverse or not, which is crucial in solving systems.
• Linear transformations: Functions that map vectors to other vectors in a way that preserves vector addition and scalar multiplication.

Linear algebra simplifies complex systems into manageable formats, making it essential in fields such as computer science, engineering, physics, and economics.

Matrix Representation

Matrix representation of a system of equations is a compact way of organizing and solving these equations using matrices. In this method, each equation is translated into a row in a matrix, with every variable represented along a column.

In the example provided, the system of equations was written as an augmented matrix:

• Row 1: Corresponds to \( x + z = 7 \), written as \([1, 0, 1, 7]\)
• Row 2: Corresponds to \( y + 2z = -6 \), written as \([0, 1, 2, -6]\)
• Row 3: Corresponds to \( 3x - y = 0 \), written as \([3, -1, 0, 0]\)

The augmented matrix includes an additional column with the constants from the equations' right side, setting up for methods like Gaussian elimination or using the matrix to find a reduced form for solutions.

This approach is systematic and can easily be performed with computers, allowing solutions to be found even for large systems of equations.