Problem 11 - Algebra and Trigonometry

Question

Write the system of linear equations represented by the augmented matrix. Use \(x, y,\) and \(z,\) or, if necessary, \(w, x, y\) and \(z,\) for the variables. $$ \left[\begin{array}{rrrr|r} {1} & {1} & {4} & {1} & {3} \\ {-1} & {1} & {-1} & {0} & {7} \\ {2} & {0} & {0} & {5} & {11} \\ {0} & {0} & {12} & {4} & {5} \end{array}\right] $$

Step-by-Step Solution

Answer

The system of linear equations is \(w + x + 4y + z = 3\), \(-w + x - y = 7\), \(2w + 5z = 11\), and \(12y + 4z = 5\).

Step 1: Write the First Equation

From the first row of the matrix, the equation is \(w + x + 4y + z = 3\).

Step 2: Write the Second Equation

From the second row of the matrix, the equation is \(-w + x - y = 7\).

Step 3: Write the Third Equation

From the third row of the matrix, the equation is \(2w + 5z = 11\).

Step 4: Write the Fourth Equation

From the fourth row of the matrix, the equation is \(12y + 4z = 5\).

Key Concepts

Understanding Augmented Matrix

An augmented matrix is a handy tool used to represent and solve systems of linear equations. This matrix combines both the coefficients of the variables and the constants from the equations. In simple terms, it's like a compact version of a system of equations, allowing you to easily apply mathematical operations to solve the system.

Imagine a typical augmented matrix as a table with two sections—the left part includes the coefficients of the variables, and the right part shows the constants, separated by a vertical line. For example, the given augmented matrix

\[ \begin{array}{rrrr|r}{1} & {1} & {4} & {1} & {3} \{-1} & {1} & {-1} & {0} & {7} \{2} & {0} & {0} & {5} & {11} \{0} & {0} & {12} & {4} & {5} \end{array}\]

is used to represent four linear equations with four variables, typically written as \(w, x, y,\) and \(z\). The coefficients are laid out by rows which directly correspond to each equation in the system. The values after the vertical line represent the constants in each equation.

Recognizing Variables in Matrix Equations

When translating between an augmented matrix and a system of linear equations, identifying the variables is crucial. Variables are the symbols, such as \(w, x, y,\) and \(z\), that represent unknown values we aim to find. In matrix notation, each column on the left side of the vertical line corresponds to one of these variables.

To read the variables in the context of the given augmented matrix, follow the order of the columns for each row:

• The first column corresponds to the coefficient of \(w\)
• The second to \(x\)
• The third to \(y\)
• The fourth to \(z\)

For instance, in the first row \(1, 1, 4, 1 | 3\), the coefficients relate to the equation \(w + x + 4y + z = 3\). Recognizing this pattern allows you to reconstruct the entire set of equations from the matrix.

Step-by-Step Solution Approach

Solving a system of linear equations represented by an augmented matrix involves carefully converting each row into an equation. Each row provides the coefficient for the respective variable in that particular equation. Let's break down how this process is executed step-by-step:

• Step 1: Focus on the first row; convert it into an equation. For example, the first row \(1, 1, 4, 1 | 3\) becomes \(w + x + 4y + z = 3\).
• Step 2: Move to the second row; this translates into the equation \(-w + x - y = 7\).
• Step 3: Use the third row; write it down as the equation \(2w + 5z = 11\).
• Step 4: Finally, the fourth row forms the equation \(12y + 4z = 5\).

Through this step-by-step approach, you can systematically derive each equation from the matrix. This method helps ensure accuracy and clarity when solving system equations, making it a powerful strategy for beginners and advanced learners alike.