Problem 13 - College Algebra

Question

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rrr|r}{3} & {2} & {0} & {3} \\ {-1} & {-9} & {4} & {-1} \\\ {8} & {5} & {7} & {8}\end{array}\right] $$

Step-by-Step Solution

Answer

The system of equations is:\[\begin{align*} 3x + 2y & = 3\\ -x - 9y + 4z & = -1\\ 8x + 5y + 7z & = 8 \\ \end{align*}\]

Step 1: Understand the Matrix Layout

The given matrix is an augmented matrix, which represents a system of linear equations. The first three columns represent the coefficients of the variables in the system, while the fourth column represents the constants on the right-hand side of each equation.

Step 2: Define Variables

Since the matrix has three columns before the vertical line, we can assume three variables. Let's label them as \(x\), \(y\), and \(z\).

Step 3: Write the First Equation

The first row of the matrix gives us the coefficients for the first equation: \(3x + 2y + 0z = 3\). Simplifying gives \(3x + 2y = 3\).

Step 4: Write the Second Equation

The second row provides the coefficients for the second equation: \(-1x - 9y + 4z = -1\), which simplifies to \(-x - 9y + 4z = -1\).

Step 5: Write the Third Equation

The third row corresponds to the third equation: \(8x + 5y + 7z = 8\).

Key Concepts

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. Instead of writing out each equation with variables explicitly, the augmented matrix keeps everything organized in rows and columns. Each row corresponds to an equation, and each column (except the last one) corresponds to the coefficients of one variable across all equations. The last column stands out as it holds the constants from the right-hand side of each equation. This neatly encapsulates the entire system in a readily workable format that is convenient for methods like Gaussian elimination.

Matrix Layout

The layout of a matrix is crucial for understanding the information it conveys. In the context of an augmented matrix:

• The columns to the left of the vertical line contain the coefficients of the variables.
• The column to the right of the vertical line contains the constants, essentially acting as the equal sign in a system of equations.

Understanding this layout helps in translating the matrix back into a set of linear equations. Each row of the matrix directly corresponds to one equation in the system, with the elements of the row representing the constants and coefficients of that equation.

Variables Definition

Defining variables is a critical early step when dealing with systems of linear equations. With three columns of coefficients before the vertical line, we generally assign three variables to our equations.

• Commonly, these variables are denoted as \(x\), \(y\), and \(z\).
• This assumption aligns with the usual convention in algebra, simplifying the notation.

Once the variables are defined, you can substitute them into each equation represented in the matrix, translating matrix data into algebraic expressions that are easier to interpret and solve.

Linear Equations Coefficients

In a system of linear equations, coefficients are the numbers that multiply the variables. They are crucial as they determine the weight or influence each variable has within its equation.

• The first column of coefficients usually multiplies the first variable (\(x\)), the second column multiplies the second variable (\(y\)), and the third column multiplies the third variable (\(z\)).
• Knowing the coefficients allows you to construct the equations fully, reflecting real-world relationships mathematically.

For example, from our matrix:

• The first row, \([3, 2, 0]\), translates to the equation: \(3x + 2y + 0z = 3\).
• Recognizing these patterns helps verify and solve the system of equations accurately.