Problem 11 - Algebra 2

Question

Write a system of equations for each augmented matrix. $$ \left[\begin{array}{rrr|r}{2} & {1} & {1} & {1} \\ {1} & {1} & {1} & {2} \\\ {1} & {-1} & {1} & {-2}\end{array}\right] $$

Step-by-Step Solution

Answer

The system of equations is \(2x + y + z = 1\), \(x + y + z = 2\), and \(x - y + z = -2\).

Step 1: Understand the Augmented Matrix

An augmented matrix represents a system of linear equations. Each row corresponds to an equation, and each column corresponds to a coefficient of a variable, with the last column representing the constants after the equality sign.

Step 2: Extract Equations from the First Row

The first row of the matrix \(\left[\begin{array}{rrr|r}{2} & {1} & {1} & {1} \end{array}\right]\) indicates the equation \(2x + y + z = 1\).

Step 3: Extract Equations from the Second Row

The second row of the matrix \(\left[\begin{array}{rrr|r}{1} & {1} & {1} & {2} \end{array}\right]\) indicates the equation \(x + y + z = 2\).

Step 4: Extract Equations from the Third Row

The third row of the matrix \(\left[\begin{array}{rrr|r}{1} & {-1} & {1} & {-2} \end{array}\right]\) indicates the equation \(x - y + z = -2\).

Key Concepts

Linear Equations

Understanding linear equations is fundamental in algebra. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form of \(ax + by + cz = d\) where \(a\), \(b\), and \(c\) are coefficients and \(d\) is the constant. They graph as straight lines when plotted in two dimensions.

In the given exercise, students were presented with an augmented matrix and asked to derive linear equations. Each row in the matrix translates into a separate linear equation. For example, the first row \(2, 1, 1 | 1\) should be read as \(2x + y + z = 1\). These exercises are designed to help students connect the abstract matrix form with practical algebraic equations and enhance their understanding of how to interpret and solve systems of linear equations.

Matrix Representation

Matrix representation offers a compact and organized way to handle systems of linear equations. A matrix is a rectangular array of numbers arranged into rows and columns. Matrices streamline operations on systems of equations, such as addition, subtraction, and multiplication, and are especially useful in solving equations using methods like elimination and substitution.

The provided exercise introduces the augmented matrix, which is a matrix with an extra column for the constants on the right side of the equations. In an augmented matrix, if you have \(n\) variables, there will be \(n + 1\) columns: \(n\) for the coefficients of the variables and one for the constants. Understanding matrix representation is key to grasping more complex concepts in algebra, such as determinant calculation, inverse matrices, and other aspects of vector spaces and linear transformations. For easier interpretation, it's important to learn how to convert the matrix back to a set of linear equations, as seen in the solution to the exercise.

Algebraic Equations

Algebraic equations are equations involving variables, coefficients, and constants, and they represent the relationship between these entities. They range from simple linear equations to more complicated polynomial, exponential, and logarithmic equations. The solving of algebraic equations involves finding the values of the variables that make the equation true.

When we talk about systems of algebraic equations, we refer to multiple equations that need to be solved simultaneously. The augmented matrix from the exercise is an example of how such systems can be represented and tackled collectively. The three extracted equations from the exercise \(\begin{array}{c}2x + y + z = 1\ x + y + z = 2\ x - y + z = -2\end{array}\) form a system that can intersect at a single point in three-dimensional space, representing a unique solution to the system. Solving algebraic equations develops analytical and problem-solving skills essential not just in mathematics, but also in fields that use mathematical modeling and data analysis.