Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the system of equations corresponding to each augmented matrix. \(\left[\begin{array}{rr|r}3 & 2 & -4 \\ 1 & -1 & 5\end{array}\right]\)

Step-by-Step Solution

Answer

The system of equations corresponding to the augmented matrix is: \(\begin{cases} 3x + 2y = -4 \\ x - y = 5 \end{cases}\)

Step 1: Identify the coefficients for each equation

The given matrix is an augmented matrix in the form of: \[ \left[\begin{array}{rr|r} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{array}\right] \] We can see that: \(a_1 = 3\), \(b_1 = 2\), \(c_1 = -4\) \(a_2 = 1\), \(b_2 = -1\), \(c_2 = 5\)

Step 2: Write the equations based on the coefficients and constants

Now, we form the system of equations using the coefficients and constants: Equation 1: \(a_1x + b_1y = c_1 \Rightarrow 3x + 2y = -4\) Equation 2: \(a_2x + b_2y = c_2 \Rightarrow x - y = 5\) So, the system of equations that corresponds to the given augmented matrix is: \(\begin{cases} 3x + 2y = -4 \\ x - y = 5 \end{cases}\)

Key Concepts

Understanding Systems of Equations

A system of equations consists of two or more equations that share common variables and are solved simultaneously. To find a solution to a system, you’re essentially looking for the intersection point(s) where both equations are true at the same time. In our example, the system has two variables, \(x\) and \(y\), and each equation in the system provides a constraint on the possible values of these variables.

By interpreting an augmented matrix, we can represent this system compactly, allowing us to manipulate it easily using various linear algebra techniques, such as row reduction. When a solution exists, it yields the value(s) of \(x\) and \(y\) that satisfy all equations in the system simultaneously. This is a fundamental concept in linear algebra and important in many applications such as engineering, physics, and computer science. Understanding how to form systems of equations from an augmented matrix is a valuable skill that serves as the basis for more complex linear algebra methods.

Deciphering Matrix Coefficients

Matrix coefficients are essentially the numerical factors that multiply the variables in a system of linear equations. In the context of an augmented matrix, these coefficients are neatly arranged in a grid format, separated from the constants (the values on the right-hand side) by a vertical line. It’s crucial to accurately identify these coefficients as they will dictate how you construct your equations.

The pattern in an augmented matrix for a system of equations generally looks like:
\[\left[\begin{array}{rr|r}a_1 & b_1 & c_1 \a_2 & b_2 & c_2\end{array}\right]\]
where \(a_i\) and \(b_i\) are the matrix coefficients for the \(i^{th}\) equation, and \(c_i\) are the constant terms. In our example, the matrix coefficients \(3, 2, 1\), and \(-1\) allow us to rebuild the original equations, setting the stage for finding their solution. Misinterpretation of these coefficients could lead to incorrect equations, so precision is key.

The Role of Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrix operations, and systems of linear equations. It provides a powerful framework for solving various problems in mathematics and applied sciences by converting them into a language of linear equations and matrices.

An augmented matrix is a key component in linear algebra; it serves as a bridge between algebraic equations and matrix operations. By converting systems of equations into matrix form, we can use a suite of techniques like Gaussian elimination or matrix inversion to find solutions more efficiently. In our exercise example, the augmented matrix represents a system of linear equations, a stepping stone to applying these advanced linear algebra tools.

Moreover, linear algebra allows us to understand geometric concepts algebraically and solve problems that would otherwise be intractable with basic algebra. This field is fundamental to modern science and technology, underpinning disciplines like computer graphics, quantum mechanics, and machine learning.