Question

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{l} 2 x-y=3 \\ 5 x+7 y=12 \end{array}\right. $$

Short answer

The augmented matrix representation of the system of equations is: \[\begin{bmatrix} 2 & -1 & | & 3 \\ 5 & 7 & | & 12 \\ \end{bmatrix}\]

Step-by-step solution

Identify coefficients and constants

The first step is identifying the coefficients and constants in the given equations. In the equation \(2x - y = 3\), 2 and -1 are coefficients of \(x\) and \(y\) respectively, while 3 is the constant. Similarly, in the equation \(5x + 7y = 12\), 5 and 7 are coefficients of \(x\) and \(y\) respectively, and 12 is the constant.

Arrange into Augmented Matrix Format

Next, arrange these coefficients and constants into the augmented matrix format. Each row is reserved for one equation. So, the first row of the matrix would be [2, -1 | 3] and the second row would be [5, 7 | 12]. The vertical bar represents the equal sign in the equations.

Key concepts

System of Linear Equations

A system of linear equations consists of two or more equations that share the same set of variables. These equations are used to find the values of the variables that satisfy all the given equations simultaneously. In this context, the system given is:\[\begin{align*} 2x - y &= 3 \ 5x + 7y &= 12 \end{align*}\]- Each equation represents a line in two-dimensional space. - The solution to the system corresponds to the point of intersection of these lines.To solve this system, you need to find the values of \(x\) and \(y\) that make both equations true at the same time. Systems like this are often solved using methods such as substitution, elimination, or graphing. However, using a matrix approach can simplify the process, especially for more complex systems.

Coefficients and Constants

Coefficients and constants are essential components of linear equations that help in their representation and solution. - **Coefficients** are the numerical factors that multiply the variables. In the equation \(2x - y = 3\), the coefficients are 2 for \(x\) and -1 for \(y\).- **Constants** are the numbers on the right side of the equation that represent the result of the equation. For \(2x - y = 3\), the constant is 3.It’s crucial to correctly identify these as they directly impact the transformation of a system of equations into its corresponding matrix form. Keeping track of these numbers will help in setting up the matrix accurately for further calculations.

Matrix Representation

Matrix representation is a powerful tool for solving systems of linear equations. It involves organizing the coefficients and constants from the system of equations into a single matrix.- In the augmented matrix, each row corresponds to an equation.- The columns are divided into two sets: one for the coefficients of the variables and another for the constants.For the given system:- The first equation \(2x - y = 3\) becomes the row \([2, -1 | 3]\).- The second equation \(5x + 7y = 12\) becomes \([5, 7 | 12]\).Thus, the augmented matrix becomes:\[\begin{bmatrix}2 & -1 & | & 3 \5 & 7 & | & 12\end{bmatrix}\]This matrix format helps streamline solving the equations using techniques like Gaussian elimination, making complex problems more manageable.