Problem 10 - College Algebra

Question

Write the augmented matrix for the linear system. \(\begin{aligned} 10.6 x+12 y+16 z &=4 \\ 19 x-5 y+3 z &=-9 \\ x+2 y &=-8 \end{aligned}\)

Step-by-Step Solution

Answer

\(\begin{bmatrix}10.6 & 12 & 16 & | & 4 \\19 & -5 & 3 & | & -9 \\1 & 2 & 0 & | & -8 \end{bmatrix}\)

Step 1: Write the coefficients

Determine the coefficients of each variable in the equations. The first equation has coefficients 10.6 for \(x\), 12 for \(y\), and 16 for \(z\). The second equation has coefficients 19 for \(x\), -5 for \(y\), and 3 for \(z\). The third equation has coefficients 1 for \(x\), 2 for \(y\), and 0 for \(z\) as there is no \(z\) term.

Step 2: Determine the constants

Identify the constants on the right-hand side of each equation. They are 4 for the first equation, -9 for the second equation, and -8 for the third equation.

Step 3: Form the augmented matrix

Using the coefficients and constants, form the augmented matrix. The rows of the matrix correspond to the equations, with the coefficients on the left and the constants in the augmented column on the right: \[\begin{bmatrix}10.6 & 12 & 16 & | & 4 \19 & -5 & 3 & | & -9 \1 & 2 & 0 & | & -8 \end{bmatrix}\]

Key Concepts

Understanding Linear Equations

A linear equation is a type of equation that involves only linear terms. Linear terms mean that each variable is raised to the power of one. The general form of a linear equation in standard textbook algebra is \( ax + by + cz + ... = d \), where:

• \( a, b, c, \) etc. are the coefficients of the variables \( x, y, z, \), respectively,
• \( d \) is a constant

In the given system, each equation includes terms that combine a read number coefficient with a variable. Linear systems form the foundation for more complicated mathematical concepts, providing a useful and simple structure for equations.

The Matrix Representation of Equations

A matrix representation is a powerful mathematical tool that helps in organizing and solving systems of equations. By using matrices, you can systematically interpret groups of equations, making them easier to solve or manipulate. In a matrix, the rows often represent each individual equation of the system, while the columns represent the different variable coefficients. The last column is reserved for the constants. In this augmented matrix, the bar (|) separates the coefficients from the constants. It highlights the equal sign in the original equations. This setup provides a structured view of our original system of equations, allowing solver operations such as row reduction or Gaussian elimination for finding solutions.

Exploring System of Equations

A system of equations consists of multiple equations that share variables. The goal is to find a set of values that satisfy all equations simultaneously. These systems can be linear or non-linear, but our focus here is on linear ones. For example, in the presented system:

• \( 10.6x + 12y + 16z = 4 \),
• \( 19x - 5y + 3z = -9 \),
• \( x + 2y = -8 \),

This set of equations needs a common solution for \( x, y, \) and \( z \).Solving these systems might involve substitution, elimination, or matrix methods, all aimed at finding that common solution.

Coefficients and Constants in Equations

Coefficients and constants play a crucial role in defining the behavior of linear equations. Coefficients are the numbers that multiply the variables. These numbers establish the proportion and rate of change of the variable they accompany. In our system:

• Equation one has coefficients \(10.6, 12, 16\)
• Equation two has \(19, -5, 3\)
• Equation three has \(1, 2, 0\)

Constants, on the other hand, are standalone numbers without variables, shifted to the right side of the equation. These constants set the specific values for each equation's result:

• The first equation has a constant of \(4\)
• The second has \(-9\)
• The third has \(-8\)

Recognizing these elements within equations helps in setting up matrices and eventually finding solutions.