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

Question

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} x+2 y &=7 \\ 2 x-y &=4 \end{aligned}\)

Step-by-Step Solution

Answer

The system of linear equations has one and only one solution, which is \((x, y) = (3, 2)\).

Step 1: Calculate the determinant of the coefficient matrix

First, let's form the coefficient matrix from the given system of linear equations: \(A = \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}\) Now, compute the determinant: \[\det(A) = (1)(-1) - (2)(2) = -1 + (-4) = -5\] Since the determinant is not equal to zero, the system has a unique solution.

Step 2: Solve the system using substitution or elimination methods

We can use either substitution or elimination methods to solve the system of equations. For this example, let's use the elimination method. First, rewrite the second equation as: \(y = 2x - 4\) Now substitute this expression for \(y\) in the first equation: \(x + 2 (2x - 4) = 7\) Simplify the equation: \(x + 4x - 8 = 7\) Now solve for \(x\): \(5x = 15\) \(x = 3\) Now that we have a value for \(x\), we can substitute it into the expression for \(y\) we found earlier: \(y = 2(3) - 4\) \(y = 6 - 4\) \(y = 2\)

Step 3: Present the solution

Based on our calculations, we found that this system of linear equations has one and only one solution, which is: \((x, y) = (3, 2)\)

Key Concepts

Unique Solution

In the system of linear equations, a unique solution exists when the equations intersect at exactly one point. This means there is a single pair of values for the variables that satisfies all the equations simultaneously.
For example, in the system given, we found the solution

• \((x, y) = (3, 2)\)

that makes both equations true. If the determinant of the coefficient matrix is not zero, the system has a unique solution. Thus, each variable can be solved exactly once, providing complete and distinct answers.

Determinant of a Matrix

The determinant of a matrix is a special number calculated from its elements. It provides important information about a system of equations.
For a 2x2 matrix \(A\) represented by

• \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\)

, the determinant is calculated as:

• \(\det(A) = ad - bc\)

In our case:
\(det(A) = (1)(-1) - (2)(2) = -1 - 4 = -5\)
Since the determinant is not zero, it confirms the uniqueness of the system’s solution.

Elimination Method

The elimination method is one of the strategies used to solve systems of equations. By adding, subtracting, or multiplying equations, the goal is to eliminate one variable, making it easier to solve for the others.
In the given solution, we derived an expression for \(y\) and substituted it into the other equation, effectively removing one variable to make calculation straightforward.
This approach can be particularly beneficial when dealing with equations that easily transform to isolate a variable. While powerful, you need to ensure that equations are manipulated correctly to maintain their balance.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that solution into the other equation. This technique allows finding the values of variables one by one.
For our system:
We solved the second equation for \(y\):
\(y = 2x - 4\)
Using this expression, we substituted \(y\) in the first equation:
\(x + 2(2x - 4) = 7\).
This simplification allows solving for \(x\) first, and then finding \(y\), breaking down the complexity into manageable steps. It's particularly efficient when one of the equations is easily solvable for a single variable.

Coefficient Matrix

The coefficient matrix in a system of linear equations is a matrix composed of the coefficients of the variables. It forms the core calculation of finding a solution to the system.
For our system, the coefficient matrix \(A\) is
\(\begin{bmatrix} 1 & 2 \ 2 & -1 \end{bmatrix}\).
This matrix can be used in different methods like calculating the determinant, which helps determine if a unique solution exists.
Understanding the arrangement and impact of coefficients in this matrix greatly aids in interpreting the system's behavior, as it directly influences the outcome of the equations' solutions.