Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{r} \frac{4}{3} x-\frac{3}{2} y=\frac{3}{4} \\ -\frac{1}{4} x+\frac{1}{3} y=\frac{2}{3} \end{array}\right. $$

Short answer

The augmented matrix is \[\left[ \begin{array}{cc|c} \frac{4}{3} & -\frac{3}{2} & \frac{3}{4} \ -\frac{1}{4} & \frac{1}{3} & \frac{2}{3} \end{array}\right] \]

Step-by-step solution

Identify the Coefficients

First, identify the coefficients of the variables and the constants from the system of equations:\(\frac{4}{3}x - \frac{3}{2}y = \frac{3}{4}\)\(-\frac{1}{4}x + \frac{1}{3}y = \frac{2}{3}\)

Write the Coefficients and Constants

Extract the coefficients and constants from the equations:Equation 1: Coefficients are \(\frac{4}{3}\), \(-\frac{3}{2}\); Constant is \(\frac{3}{4}\)Equation 2: Coefficients are \(-\frac{1}{4}\), \(\frac{1}{3}\); Constant is \(\frac{2}{3}\)

Form the Augmented Matrix

Create the augmented matrix using the coefficients and constants identified:\[\left[ \begin{array}{cc|c} \frac{4}{3} & -\frac{3}{2} & \frac{3}{4} \ -\frac{1}{4} & \frac{1}{3} & \frac{2}{3} \end{array}\right] \]

Key concepts

System of Equations

A system of equations is a set of two or more equations with the same set of variables. The main goal in solving a system of equations is to find the value(s) of the variables that satisfy all the equations simultaneously.
In our example, we have the following system of equations:
\(\frac{4}{3} x - \frac{3}{2} y = \frac{3}{4}\) and \(-\frac{1}{4} x + \frac{1}{3} y = \frac{2}{3}\).
To solve this, we need to find values of x and y that make both equations true at the same time.

• The first equation says that \(\frac{4}{3} x\) minus \(\frac{3}{2} y\) equals \(\frac{3}{4}\).
• The second equation tells us that \(-\frac{1}{4} x\) plus \(1/3 y\) equals \(2/3\).

The process of solving this system can be made simpler by representing it in a different form, such as an augmented matrix.

Coefficients

Coefficients are the numerical factors that multiply the variables in an equation.
In our system of equations, the coefficients are the numbers in front of the variables x and y.
Let's break down the coefficients for our equations:

• In \(\frac{4}{3} x - \frac{3}{2} y = \frac{3}{4}\), the coefficients are \(\frac{4}{3}\) for x and \(-\frac{3}{2}\) for y.
• In \(-\frac{1}{4} x + \frac{1}{3} y = \frac{2}{3}\), the coefficients are \(-\frac{1}{4}\) for x and \(\frac{1}{3}\) for y.

To write the augmented matrix, we first need to identify these coefficients accurately.

Constants

Constants in a system of equations are the standalone numbers without attached variables.
They are typically on the right side of the equations.
In the system of equations given: \(\frac{4}{3} x - \frac{3}{2} y = \frac{3}{4}\) and \(-\frac{1}{4} x + \frac{1}{3} y = \frac{2}{3}\), the constants are the values on the right side of the equations: \(\frac{3}{4}\) and \(\frac{2}{3}\).
Identifying constants is crucial for setting up the augmented matrix correctly because they are placed in the last column of this matrix.

Matrix Representation

A matrix representation helps simplify systems of equations, making them easier to solve using different mathematical operations.
An augmented matrix combines the coefficients and constants of a system of equations into a structured grid.
For our system: \(\frac{4}{3} x - \frac{3}{2} y = \frac{3}{4}\) and \(-\frac{1}{4} x + \frac{1}{3} y = \frac{2}{3}\), we have:
Coefficients of the first equation: \(\frac{4}{3}\), \(-\frac{3}{2}\)
Coefficients of the second equation: \(-\frac{1}{4}\), \(\frac{1}{3}\)
Constants: \(\frac{3}{4}\), \(\frac{2}{3}\)
This information can be organized into an augmented matrix: \[\begin{array}{cc|c} \frac{4}{3} & -\frac{3}{2} & \frac{3}{4} \ -\frac{1}{4} & \frac{1}{3} & \frac{2}{3} \ \end{array}\]
Augmented matrices are useful because they allow us to use operations like row reduction to find solutions more easily.