Problem 8 - Contemporary Precalculus

Question

In Exercises \(5-8,\) the augmented matrix of a system of equations is given. Express the system in equation notation. $$\left(\begin{array}{rrrr} -1 & 0 & 2 & 6 \\ 0 & 5 & -4 & 1 \\ 8 & -2 & 3 & 4 \end{array}\right)$$

Step-by-Step Solution

Answer

Question: Convert the given augmented matrix into a system of equations. Augmented Matrix: \(\begin{bmatrix} -1 & 0 & 2 & 6 \\ 0 & 5 & -4 & 1 \\ 8 & -2 & 3 & 4 \end{bmatrix}\) Answer: The system of equations corresponding to the given augmented matrix is: \begin{cases} -x + 2z = 6\\ 5y - 4z = 1\\ 8x - 2y + 3z = 4 \end{cases}

Step 1: Identify the variables in the system

Since there are three non-augmented columns in the given matrix, the system consists of three variables, which we will denote as \(x, y,\) and \(z\).

Step 2: Write the first equation from the first row

The first row is \((-1, 0, 2, 6)\). This corresponds to the equation \(-1x + 0y + 2z = 6\). We can simplify this to \(-x + 2z = 6\).

Step 3: Write the second equation from the second row

The second row is \((0, 5, -4, 1)\). This corresponds to the equation \(0x + 5y - 4z = 1\). We can simplify this to \(5y - 4z = 1\).

Step 4: Write the third equation from the third row

The third row is \((8, -2, 3, 4)\). This corresponds to the equation \(8x - 2y + 3z = 4\).

Step 5: Write the system of equations

Now that we have the three equations, we can write the system of equations: \begin{cases} -x + 2z = 6\\ 5y - 4z = 1\\ 8x - 2y + 3z = 4 \end{cases}

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool used to represent systems of equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. This format is especially useful when applying methods like Gaussian elimination for solving systems of equations.

• The matrix contains a vertical bar that separates the matrix into two parts: the coefficients of the variables make up the left part, and the constants from the equations form the right part, known as the augment.
• Each row corresponds to one equation in the system.
• The number of columns before the vertical bar corresponds to the number of variables in the system.

For example, the given augmented matrix: \[\begin{pmatrix} -1 & 0 & 2 & \vert & 6 \0 & 5 & -4 & \vert & 1 \8 & -2 & 3 & \vert & 4 \end{pmatrix}\] indicates a system with three variables. Each row translates into an equation by associating the coefficients with their respective variables and setting them equal to the constants in the last column.

Equation Notation

Equation notation is a mathematical language used to express relationships between different quantities using variables, numbers, and operations. Writing a system of equations from an augmented matrix requires translating the matrix rows into equations. By doing so, you will better understand how the variables interact in your system.

• Start by identifying the number of variables from the number of columns before the augment in the matrix.
• Each row of the matrix represents an equation. For instance, the row \(-1, 0, 2, 6\) translates to the equation \(-x + 2z = 6\).
• The absence of a variable's coefficient in an equation implies a coefficient of zero.

For the augmented matrix example, using the systematic translation process results in the system: \[\begin{cases}-x + 2z = 6\5y - 4z = 1\8x - 2y + 3z = 4\end{cases}\] This notation clearly expresses each equation in terms of its variables and constant values.

Variables in Equations

Variables are symbolic representations that stand in for unknown values in equations. They are the building blocks of a system of equations and help us understand what we are solving for. Recognizing and working with variables is a key step in dealing with systems of equations.

• In a system, the variables typically represented by letters such as \(x, y, z\), correspond to the non-augmented columns in an augmented matrix.
• Each variable can assume infinite possible values, but the system aims to find specific sets of values that satisfy all the equations simultaneously.
• No variable appears in isolation; instead, it collaborates with other variables through mathematical expressions involving addition, subtraction, and coefficients.

For example, in the system derived from our matrix: \[\begin{cases}-x + 2z = 6\5y - 4z = 1\8x - 2y + 3z = 4\end{cases}\] we have three variables, \(x, y, z\). Solving the system means finding specific values of \(x, y, and z\) that satisfy all three equations simultaneously, revealing the relationship between them.