Problem 35 - College Algebra

Question

Write each matrix equation as a system of linear equations without matrices. $$\left[\begin{array}{rrr}2 & 0 & -1 \\\0 & 3 & 0 \\\1 & 1 & 0\end{array}\right]\left[\begin{array}{l} x \\\y \\\z\end{array}\right]=\left[\begin{array}{l}6 \\\9 \\\5\end{array}\right]$$

Step-by-Step Solution

Answer

\[2x - z = 6, 3y = 9, x + y = 5\]

Step 1: Formulate first equation

From the first row of the matrix, form a linear equation by multiplying: \(2x + 0y -1z = 6\). This simplifies to: \(2x - z = 6\).

Step 2: Formulate second equation

From the second row of the matrix, a linear equation can be formed by multiplying: \(0x + 3y + 0z = 9\). This simplifies to: \(3y = 9\).

Step 3: Formulate third equation

From the third row of the matrix, a linear equation can be formed by multiplying: \(x + y + 0z = 5\). This simplifies to: \(x + y = 5\).

Key Concepts

Matrix Equation

A matrix equation is a compact way of representing multiple linear equations. It uses matrices and vectors to simplify the presentation of a system of equations. Suppose you have a matrix \( A \) and a vector \( \mathbf{x} \), the product \( A\mathbf{x} \) results in another vector. This vector is equivalent to a series of linear combinations of \( \mathbf{x} \). This combination translates into separate linear equations. In our exercise, the equation \(\left[\begin{array}{rrr}2 & 0 & -1 \0 & 3 & 0 \1 & 1 & 0\end{array}\right]\left[\begin{array}{l} x \y \z\end{array}\right]=\left[\begin{array}{l}6 \9 \5\end{array}\right]\) can be split into separate linear equations.

• The first row results in the equation \(2x - z = 6\)
• The second row gives \(3y = 9\)
• The third row translates to \(x + y = 5\)

Understanding how to convert matrix equations to linear equations is fundamental in matrix algebra and system solving.

Linear Algebra

Linear algebra is a vibrant branch of mathematics that focuses on vectors, matrices, and linear mappings between spaces. It provides the groundwork for many areas such as computer science, physics, and engineering. This field allows us to handle a set of equations using matrices which simplifies computations greatly. Key Elements of Linear Algebra:

• Matrices: Rectangular arrays of numbers or expressions.
• Vectors: Lists of numbers that can represent points, directions, or other quantities.
• Linear transformations: Functions mapping vectors to vectors following this rule: \(T(ax + by) = aT(x) + bT(y)\).

Linear algebra acts like a toolkit that offers powerful methods for dealing with complex systems of equations. It allows problems to be structured in convenient forms for solutions.

Solving Systems of Equations

Solving systems of linear equations involves finding values for variables that satisfy all equations simultaneously. This can be approached using several methods, and here are a few: Methods to Solve Systems:

• Substitution: Solve one equation for one variable, then substitute into another equation.
• Elimination: Add or subtract equations to eliminate variables step-by-step.
• Matrix Method: Use row operations on matrices to reach solutions efficiently.

For systems expressed in matrix form, the matrix method is often unbeatable in efficiency and ease. Each technique, however, shares the common goal of simplifying the problem down into manageable pieces until a straightforward solution path emerges. By mastering these methods, one becomes proficient at dissecting and understanding the dynamics of systems of equations.