Question

Convert the given system of linear equations into an augmented matrix. $$ \begin{array}{l} 2 x+5 y-6 z=2 \\ 9 x\quad-8 z=10 \\ -2 x+4 y+z=-7 \end{array} $$

Short answer

The augmented matrix is \(\begin{bmatrix} 2 & 5 & -6 & | & 2 \\ 9 & 0 & -8 & | & 10 \\ -2 & 4 & 1 & | & -7 \end{bmatrix}\).

Step-by-step solution

Write the System in Standard Form

Make sure each equation in the system is written in standard form, aligning like terms vertically. The given system is: - Equation 1: \(2x + 5y - 6z = 2\) - Equation 2: \(9x + 0y - 8z = 10\) (note the placeholder zero for the \(y\) term) - Equation 3: \(-2x + 4y + z = -7\)

Identify Coefficients and Constants

For each equation, identify the coefficients of \(x\), \(y\), and \(z\), as well as the constant term on the right-hand side. - Equation 1 coefficients: \(2, 5, -6\), and constant \(2\) - Equation 2 coefficients: \(9, 0, -8\), and constant \(10\) - Equation 3 coefficients: \(-2, 4, 1\), and constant \(-7\)

Construct the Augmented Matrix

Using the coefficients from each equation, construct the augmented matrix. Place the coefficients in the square matrix part and the constants on the right side after the vertical line. The resulting augmented matrix is: \[\begin{bmatrix}2 & 5 & -6 & | & 2 \9 & 0 & -8 & | & 10 \-2 & 4 & 1 & | & -7\end{bmatrix}\]

Key concepts

System of Linear Equations

A system of linear equations is a collection of two or more equations with the same set of variables. In these equations, each variable typically represents an unknown value we aim to solve.
These systems can be solved using various methods like substitution, elimination, or matrix methods. When dealing with such systems:

• Each equation provides a line in the coordinate plane (or a hyperplane in higher dimensions).
• The solution of the system is the point(s) where these lines intersect.

Understanding how to manipulate and solve these systems is essential for mathematicians, scientists, and engineers. They are foundational in fields like algebra and calculus.

Coefficients and Constants

In a system of linear equations, each variable is multiplied by a number. These numbers are called coefficients. They provide critical information about the equation because they describe how each variable affects the outcome of the equation.

• For example, in the equation \( 2x + 5y - 6z = 2 \), the coefficients are 2, 5, and -6 for the variables \(x\), \(y\), and \(z\), respectively.
• The constant is the number on the right side of the equation, such as 2 in the equation above.

This constant often represents the solution corresponding to each combination of variables that satisfies the equation.

Standard Form

Writing an equation in standard form means organizing it neatly, aligning all terms, so they are easy to compare and handle in systems. Each equation should have all the variable terms on one side and the constant on the other.

• A typical standard form is \( Ax + By + Cz = D \), where \(A\), \(B\), and \(C\) are coefficients, and \(D\) is the constant.

Ensuring all equations are in this form is crucial when converting them into other formats, such as matrices. It helps to maintain clarity and ease any further calculations.

Matrix Representation

Matrices are a powerful tool in mathematics to represent and solve systems of linear equations efficiently. When a system of equations is represented as a matrix, it simplifies operations like solving and manipulating the equations. To convert a system into its matrix form:

• The coefficients of the equations are organized into rows and columns, forming a matrix.
• An augmented matrix includes a column for the constants, separated by a delimiter like a vertical line to differentiate from the coefficients.
• For the given system, the augmented matrix is represented as: \[\begin{bmatrix}2 & 5 & -6 & | & 2 \9 & 0 & -8 & | & 10 \-2 & 4 & 1 & | & -7\end{bmatrix}\]

By using matrices, we streamline the approach to solving complex systems, making it easier to apply computational methods like the Gauss-Jordan elimination or using software tools for larger systems.