Problem 12 - Precalculus with Limits

Question

In Exercises 7-16, use Cramer's Rule to solve (if possible) the system of equations. \(\begin{cases} 2.4x - 1.3y = 14.63 \\ -4.6x + 0.5y = -11.51 \end{cases}\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1.60\) and \(y = 19.85\).

Step 1: Calculate the determinant of the coefficient matrix

First, the determinant D of the coefficient matrix is calculated. The coefficient matrix is \([2.4, -1.3; -4.6, 0.5]\), so \(D = 2.4*0.5 - (-1.3)*(-4.6) = 1.2 - 5.98 = -4.78\).

Step 2: Calculate Dx

Next, the determinant Dx is calculated. This is done by replacing the x coefficients in the original matrix with the solution coefficients, followed by calculating the determinant exactly as before. Therefore, the matrix for Dx is \([14.63, -1.3; -11.51, 0.5]\) and \(Dx = 14.63*0.5 - (-1.3)*(-11.51) = 7.315 - 14.963 = -7.648\).

Step 3: Calculate Dy

To find the determinant Dy, the y coefficients in the original matrix are replaced with the solution coefficients. Therefore, the Dy matrix becomes \([2.4, 14.63; -4.6, -11.51]\) and \(Dy = 2.4*(-11.51) - 14.63*(-4.6) = -27.624 - 67.288 = -94.912\).

Step 4: Solve for x and y

Cramer's rule states that \(x = Dx/D\) and \(y = Dy/D\), so \(x = -7.648/-4.78 = 1.60\) and \(y = -94.912/-4.78 = 19.85\)

Key Concepts

Determinant

The determinant is a special number calculated from a square matrix. It provides essential information regarding the matrix's characteristics, such as whether it is invertible.
In our exercise, we have a 2x2 matrix derived from the coefficients of the system of equations.

• Determinant for a 2x2 matrix: Given a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \\), the determinant is computed as \( ad - bc \\).
• If the determinant is zero, the system of equations may not have a unique solution.

A non-zero determinant, as in our step-by-step solution, signifies that the matrix is non-singular and the system of equations can be solved using Cramer's Rule.

System of Equations

A system of equations consists of multiple equations with multiple variables. In this case, we are working with a system of linear equations that deals with two variables, \( x \) and \( y \).
The solution to a system of equations is the set of values for the variables that make all equations true simultaneously.
In mathematical problems, these variables typically represent quantities of interest, like distances, prices, or any other measurable concept.

• Simplicity: Single unique solutions make a system consistent and indepedent.
• Graphical interpretation: For two variables, each equation can be represented as a line in a plane, and solving the system means finding the intersection point of these lines.

In our specific problem, the equations describe lines in a plane, and the solution \( (x, y) = (1.60, 19.85) \) is the point where the two lines intersect.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices and their operations. In our context, it involves setting up and manipulating matrices to solve systems of equations.
There are several operations within matrix algebra, such as addition, multiplication, and finding determinants, that facilitate problem-solving.

• Coefficient Matrix: Represents the coefficients of the variables from a system of linear equations.
• Determinants are used within these matrices to apply Cramer's Rule, among others.

Matrix algebra is extremely powerful in solving complex systems that are cumbersome to handle using traditional algebraic methods. In our exercise, we employed matrix representation and determinant calculation as part of Cramer's Rule to find a solution efficiently.