Question

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{r} x+10 y-3 z=2 \\ 5 x-3 y+4 z=0 \\ 2 x+4 y=6 \end{array}\right. $$

Short answer

The augmented matrix for the given system of linear equations is: \[\begin{bmatrix}1 & 10 & -3 & 2 \5 & -3 & 4 & 0 \2 & 4 & 0 & 6\end{bmatrix}\]

Step-by-step solution

Identify the Coefficients and Constants

Look at each equation and identify the coefficients of each variable as well as the constants on the other side of the equation. For the first equation, this is \(x\), \(10y\), \(-3z\), and \(2\). For the second equation, this is \(5x\), \(-3y\), \(4z\), and \(0\). For the third equation, it is \(2x\), \(4y\), and \(6\). Note that there is no \(z\) term in this equation.

Construct the Augmented Matrix

Create a matrix where each row represents an equation. Each coefficient should have its own spot in the row, and the constant should occupy the final spot in the row. The matrix should look like this: \[\begin{bmatrix}1 & 10 & -3 & 2 \5 & -3 & 4 & 0 \2 & 4 & 0 & 6\end{bmatrix}\]

Review

Ensure each coefficient and constant from the original equations has been correctly transferred to the augmented matrix. The absence of the \(z\) term in the third equation is represented by a \(0\) in the third row of the 'z' column.

Key concepts

Understanding Linear Equations

Linear equations are the foundation of algebra and represent relationships between variables that add up to a constant value. These equations can have one (e.g., \( x + 2 = 6 \)), two (e.g., \(2x + 3y = 6\)), or more variables and are called 'linear' because they form a straight line when graphed on a coordinate plane.

Consider a system of linear equations as a snapshot of several lines and the solution to this system as the points where these lines intersect. In the provided exercise, we're given a set of three linear equations with three variables – \(x\), \(y\), and \(z\). To solve these equations simultaneously, it's essential first to align them in a structured form to make it easier to apply operations that can lead to finding the variables' values.

Matrix Representation of a System of Equations

Matrix representation is a compact and structured way to display a system of linear equations. It involves arranging the algebraic coefficients of the variables and constants of the system into an array known as a matrix. This approach simplifies solving linear systems, especially when dealing with multiple equations and variables.

Each row in the matrix corresponds to a specific equation, and each column represents a particular variable, extending to include the equation constants as an additional column, thus we create an 'augmented matrix'. For instance, an augmented matrix for the given system of three equations would organize the coefficients and constants in a rectangular array where the last column signifies the constants from each equation, separate from the coefficients. By using certain techniques like row reduction, we can then proceed to solve for the variables in a step-by-step manner.

Algebraic Coefficients and Their Role

Algebraic coefficients are numbers that precede and multiply the variables within an equation. They construct the quantitative aspect of the relationship defined by the equation. For example, in the first equation of our given problem, \(1\) is the implicit coefficient of \(x\), and \(10\) and \(-3\) are the coefficients for \(y\) and \(z\), respectively.

When creating an augmented matrix, identifying these coefficients correctly is crucial as they define the matrix and hence the entire system's behavior. A change in a coefficient alters the corresponding lines' slopes, and, as a result, the system's solution set can significantly shift. Moreover, when there is no presence of a variable in any equation (like \(z\) in the third equation for the given exercise), the coefficient is considered as \(0\), illustrating its absence in the matrix format.