Problem 9 - College Algebra

Question

Write the augmented matrix for the linear system. \(\begin{aligned} \text { 9. } x+5 y+8 z &=19 \\ 12 x+3 y &=4 \\ 3 x+4 y+9 z &=-7 \end{aligned}\)

Step-by-Step Solution

Answer

The augmented matrix is: \(\begin{bmatrix} 1 & 5 & 8 & | & 19 \\ 12 & 3 & 0 & | & 4 \\ 3 & 4 & 9 & | & -7 \end{bmatrix}\).

Step 1: Arrange the Equations

Start by writing the three given equations in their standard linear form: \(x + 5y + 8z = 19\), \(12x + 3y + 0z = 4\), and \(3x + 4y + 9z = -7\). Ensure that coefficients for each variable are identified, even if a variable is missing (for instance, \(z = 0\) in the second equation).

Step 2: Identify Coefficients and Constants

For each equation, list the coefficients of \(x\), \(y\), and \(z\), followed by the constant term on the right side of the equation. The coefficients and constants for each equation are: Equation 1: \([1, 5, 8, 19]\), Equation 2: \([12, 3, 0, 4]\), Equation 3: \([3, 4, 9, -7]\).

Step 3: Form the Augmented Matrix

Combine the lists of numbers from each equation into a single matrix. This matrix includes all coefficients and constants from the equations, where each row represents an equation. The augmented matrix is: \[\begin{bmatrix} 1 & 5 & 8 & | & 19 \ 12 & 3 & 0 & | & 4 \ 3 & 4 & 9 & | & -7 \end{bmatrix}\] The vertical bar separates the coefficients of the variables from the constants.

Key Concepts

Understanding Linear Equations

Linear equations form the backbone of algebra and are equations that express relationships where variables are raised only to the first power. In essence, these equations form straight lines when graphed, hence the term 'linear.' Linear equations can have one or more variables, such as:

• One variable: e.g., \(x = 5\)
• Two variables: e.g., \(x + 2y = 10\)
• Three variables: e.g., \(x + 3y + 2z = 12\)

The key objective when solving linear equations is to find values for the variables that make all equations true simultaneously. This system of equations can be represented in various forms, with the matrix being one of the most useful representations for solving systems of equations efficiently. When dealing with multiple equations, aligning them into the standard linear form \(ax + by + cz + \,...\, = d\) is essential for consistency, making the set-up for solving much smoother.

Exploring Coefficients

Coefficients are vital components of linear equations as they indicate the proportion of each variable in an equation. In essence, these numbers sit directly in front of the variables like \(x\), \(y\), and \(z\), and they define how much each variable contributes to the expression.Whenever you write a linear equation, like \(12x + 3y + 0z = 4\), the numbers \(12\), \(3\), and \(0\) stand as coefficients. Each coefficient - Multiplies with its associated variable- Determines the steepness and direction if the equation were plottedIf a variable isn't included in an equation, its coefficient is \(0\). Recognizing coefficients helps us transform equations from a written form into a matrix representation, crucial for operations like row reduction or using techniques to find solutions. By focusing on coefficients, we essentially decode the equation’s "weight" or "influence" in the system, making problem-solving efficient and systematic.

Matrix Representation of Linear Equations

Matrices offer a compact way to represent and manipulate systems of linear equations. The augmented matrix is especially useful because it combines all the information from our equations into a neat array.

• Each row in the matrix corresponds to one equation in the system.
• Each column, except the last, represents the coefficients of the variables across different equations.
• The last column includes the constants, or right-hand sides of the equations.

For example, given our system of equations, the matrix:\[\begin{bmatrix}1 & 5 & 8 & | & 19 \12 & 3 & 0 & | & 4 \3 & 4 & 9 & | & -7 \\end{bmatrix}\]helps systematically represent each equation’s influence on the system. The vertical bar is there to differentiate between coefficients of the variables and the constants. This format facilitates techniques like Gaussian elimination or other matrix operations, enabling us to efficiently find solutions to even large-scale systems of linear equations.