Problem 35 - Contemporary Precalculus

Question

Solve the system of equations. $$\begin{aligned}&a x+b y=r \quad \text { (where } a, b, c, d, r, s \text { are }\\\&c x+d y=s \quad \text { constants and } a d-b c \neq 0)\end{aligned}$$

Step-by-Step Solution

Answer

Answer: The unique solution for the given system of linear equations can be found using Cramer's rule and is represented by the ordered pair (x, y) where x = Δx / Δ and y = Δy / Δ.

Step 1: Find the determinant of the coefficients (Δ)

Calculate the determinant of the matrix formed by the coefficients a, b, c, and d. This is represented by Δ and can be found using the formula Δ = ad - bc. Make sure ad - bc is not equal to 0, otherwise the system doesn't have a unique solution.

Step 2: Find the determinant of the x-matrix (Δx)

Replace the first column (coefficients of x) in the matrix with the constants r and s, while keeping the second column (coefficients of y) the same. This new matrix will be used to find the determinant Δx. Calculate Δx using the formula: Δx = rd - bs.

Step 3: Find the determinant of the y-matrix (Δy)

Replace the second column (coefficients of y) in the original matrix with the constants r and s, while keeping the first column (coefficients of x) the same. This new matrix will be used to find the determinant Δy. Calculate Δy using the formula: Δy = as - bc.

Step 4: Calculate the value of x

Use the values of Δ and Δx to find the value of x using the formula: x = Δx / Δ.

Step 5: Calculate the value of y

Use the values of Δ and Δy to find the value of y using the formula: y = Δy / Δ.

Step 6: Present the solution as an ordered pair

Combine the values of x and y to form an ordered pair (x, y). This ordered pair represents the unique solution for the given system of linear equations.

Key Concepts

Determinants

Understanding determinants is crucial for dealing with systems of linear equations. Essentially, a determinant is a special number calculated from a square matrix. For a 2x2 matrix, it provides information about the matrix, such as whether it has a unique solution. The formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(ad - bc\). This value of the determinant tells us a lot about the system of equations.

• If the determinant is zero, it signifies that the system of equations does not have a unique solution. It might have either no solutions or infinitely many.
• If the determinant is non-zero, you have a unique solution for the system.

For the problem at hand, the determinant \(\Delta = ad - bc\) must not be zero to ensure a unique solution.

Matrix Algebra

Matrix algebra is a powerful tool for solving linear equations, especially when dealing with multiple equations at once. Matrices offer a neat way to organize coefficients and constants from equations. In our given linear system, we can represent the coefficients as a matrix:
\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \] and the constants as a vector \( \begin{bmatrix} r \ s \end{bmatrix} \).

This allows us to use matrix operations to find solutions. Standard operations include addition, subtraction, and multiplication of matrices. These operations make it possible to manipulate entire systems of equations simultaneously. By calculating determinants and employing rules like Cramer's, we can efficiently find solutions to the system.

Cramer's Rule

Cramer's Rule provides a straightforward method for solving systems of linear equations using determinants. It's especially handy when you know the system has a unique solution. Following Cramer's Rule involves these steps:

• Compute the determinant of the coefficient matrix, \(\Delta\).
• For \(\Delta_x\), replace the first column of the original matrix with the constants \(r\) and \(s\).
• For \(\Delta_y\), replace the second column with the constants.

Then, the solutions for \(x\) and \(y\) are found as follows:
\[ x = \frac{\Delta_x}{\Delta} \]
\[ y = \frac{\Delta_y}{\Delta} \]
Cramer's Rule is elegant but is most efficient for small systems, typically those with two or three variables.

Unique Solution

The notion of a unique solution in a system of linear equations indicates that there is precisely one set of values for the variables that satisfies every equation in the system.
For the system \( ax + by = r \) and \( cx + dy = s \), finding a unique solution means solving it such that both equations are satisfied simultaneously by one specific point \((x, y)\).

• The determinant \(\Delta\) must be non-zero.
• If \(\Delta = 0\), the lines represented by the equations might be parallel (no solution) or coincident (infinite solutions).

When \(\Delta\) is non-zero, it suggests the intersection at one point, thereby guaranteeing a unique solution. In practical terms, this means we can confidently solve for \(x\) and \(y\) using the formulas derived from Cramer's Rule.