Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, write the augmented matrix for the linear system. $$ \begin{array}{l} x+5 y+8 z=19 \\ 12 x+3 y=4 \\ 3 x+4 y+9 z=-7 \end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Write the Coefficient Matrix

Identify and write down the coefficients of the variables from each equation. Align them in a matrix format. The system of equations is:\(x + 5y + 8z = 19\) \(12x + 3y = 4\) \(3x + 4y + 9z = -7\) So, the coefficient matrix is:\[\begin{bmatrix}1 & 5 & 8 \12 & 3 & 0 \3 & 4 & 9\end{bmatrix}\]

Step 2: Assemble the Constants into a Constant Column

Extract the constants from the right-hand side of each equation and form a column matrix. From the equations:- The constant for the first equation is \( 19 \).- The constant for the second equation is \( 4 \).- The constant for the third equation is \( -7 \).Thus, the constant column is:\[\begin{bmatrix}19 \4 \-7\end{bmatrix}\]

Step 3: Formulate the Augmented Matrix

Combine the coefficient matrix and the constant column to form the augmented matrix. The constant column is added as an additional column to the right of the coefficient matrix. The augmented matrix is:\[\begin{bmatrix}1 & 5 & 8 & \vert & 19 \12 & 3 & 0 & \vert & 4 \3 & 4 & 9 & \vert & -7\end{bmatrix}\]

Key Concepts

Linear Systems

A linear system consists of two or more linear equations involving the same set of variables. In the context of the given example, we have three equations with variables \( x \), \( y \), and \( z \). The goal of solving this linear system is to find the values of these variables that satisfy all equations simultaneously. These problems often arise in various practical scenarios, such as in physics or economics, where you need to find the intersection or balancing point of different constraints or conditions.

• Linear equations have the basic form of \( ax + by + cz = d \), where \( a \), \( b \), and \( c \) are coefficients representing the respective impact or contribution of the variables \( x \), \( y \), and \( z \).
• The solution to a linear system can be found using different methods, including substitution, elimination, or matrix operations.
• In matrix terms, linear systems help in transforming complex algebraic dynamics into a more structured and computational-friendly format.

Understanding linear systems is crucial as they form the foundation for more advanced topics in mathematics and applied sciences.

Coefficient Matrix

The coefficient matrix is a systematic way to organize and display the coefficients of the variables present in a set of linear equations. For the given exercise, the coefficient matrix was crafted from the coefficients found in the equations: \( x + 5y + 8z = 19 \), \( 12x + 3y = 4 \), and \( 3x + 4y + 9z = -7 \).

• To create the coefficient matrix, you extract the numbers in front of each variable across all equations.
• These numbers are aligned in a rectangular array where each row corresponds to an equation, and each column corresponds to a specific variable.

This process results in the matrix:\[\begin{bmatrix}1 & 5 & 8 \12 & 3 & 0 \3 & 4 & 9\end{bmatrix}\]This matrix eloquently encapsulates all the coefficients of \( x \), \( y \), and \( z \) respectively. Notice that a zero is used for missing terms; this keeps the matrix dimensionally consistent. Coefficient matrices are integral in representing linear systems in a form suitable for algorithmic solutions.

Constant Matrix

The constant matrix is often referred to as a column matrix representing the constants from the right side of each equation in a linear system. This matrix plays a crucial role in forming the augmented matrix and analyzing the linear system.

• These constants are simply the numbers balanced on the equation's right side, representing the solutions these linear combinations equate to.
• They are assembled into a separate column vector apart from the coefficients of the variables.

In the given example, the constants from each equation are extracted to form the following constant matrix:\[\begin{bmatrix}19 \4 \-7\end{bmatrix}\]This matrix provides information on what the linear combinations equate to, fundamentally serving as the target value or output for the given system of equations. Pairing this with the coefficient matrix can efficiently facilitate solution finding through matrix operations.

Matrix Representation

Matrix representation is a powerful mathematical tool that permits complex linear systems to be expressed in a compact and precise format, ideal for computational handling. The augmented matrix is a particular form that includes both the coefficient matrix and the constant matrix.

• This method blends the coefficients of the variables together with their corresponding constants into one comprehensive matrix.
• An augmented matrix is defined by adding the constant column to the coefficient matrix, separating the two using a vertical bar for clarity.

In the given example, the augmented matrix appears as:\[\begin{bmatrix}1 & 5 & 8 & \vert & 19 \12 & 3 & 0 & \vert & 4 \3 & 4 & 9 & \vert & -7\end{bmatrix}\]This succinct representation allows for applying matrix methodologies such as row reduction to solve the linear system efficiently. It is an indispensable approach in both theoretical and applied aspects of linear algebra, tightly integrating the ideas behind coefficient and constant matrices.