Problem 16 - Precalculus

Question

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} x-4 y=-7 \\ 3 x+8 y=19 \end{array}$$

Step-by-Step Solution

Answer

The solution is \((x, y) = (1, 2)\).

Step 1: Write the system in matrix form

The given system of equations can be written in the form of a matrix equation \ \( AX = B \) \, where \ \( A \) is the coefficient matrix, \ \( X \) is the column matrix of variables, and \ \( B \) is the column matrix of constants. In this case, \[A = \begin{bmatrix} 1 & -4 \ 3 & 8 \end{bmatrix},\ X = \begin{bmatrix} x \ y \end{bmatrix},\ B = \begin{bmatrix} -7 \ 19 \end{bmatrix}\]

Step 2: Calculate the determinant of matrix A

The determinant of matrix \ \( A \) is calculated as follows: \\[\det(A) = \begin{vmatrix} 1 & -4 \ 3 & 8 \end{vmatrix} = (1)(8) - (3)(-4) = 8 + 12 = 20\]Since the determinant is non-zero, Cramer's rule can be applied.

Step 3: Find the determinant for x - \\(\det(A_x)\\)

To find \ \( x \), replace the first column of \ \( A \) with matrix \ \( B \) and calculate the determinant. \[A_x = \begin{bmatrix} -7 & -4 \ 19 & 8 \end{bmatrix}\]Then,\[\det(A_x) = (-7)(8) - (19)(-4) = -56 + 76 = 20\]

Step 4: Find the determinant for y - \\(\det(A_y)\\)

To find \ \( y \), replace the second column of \ \( A \) with matrix \ \( B \) and calculate the determinant. \[A_y = \begin{bmatrix} 1 & -7 \ 3 & 19 \end{bmatrix}\]Then,\[\det(A_y) = (1)(19) - (3)(-7) = 19 + 21 = 40\]

Step 5: Apply Cramer's Rule

Using Cramer's Rule, the solutions for \ \( x \) and \ \( y \) are given by:\[x = \frac{\det(A_x)}{\det(A)} = \frac{20}{20} = 1\]\[y = \frac{\det(A_y)}{\det(A)} = \frac{40}{20} = 2\]

Step 6: Solution of the System

The solution to the system of equations is \ \( x = 1 \) and \ \( y = 2 \). Thus, the point \( (x, y) = (1, 2) \) satisfies both equations.

Key Concepts

Understanding Determinants

The determinant is a special number that you can calculate from a square matrix. It provides important properties of the matrix, such as whether it is invertible or not. For a 2x2 matrix like \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\] the determinant is calculated as:\[\text{det}(A) = ad - bc.\]Understanding determinants is crucial because

• If the determinant of a matrix is zero, it means that the system of equations has no unique solutions.
• If the determinant is non-zero, the system can be uniquely solved using methods like Cramer's Rule.

In the example provided, the determinant of matrix A is 20. This non-zero value confirms that Cramer's Rule can be applied to find a unique solution to the system of equations. Remember, the determinant acts as a gateway — only when it is non-zero can you proceed with solving using Cramer's Rule or finding the inverse of the matrix.

Decoding Systems of Equations

A system of equations involves finding the values of variables that satisfy all the given equations simultaneously. In our example, we have:\[\begin{array}{r}x - 4y = -7 \3x + 8y = 19\end{array}\]These two linear equations form a system that we need to solve. Here's why understanding systems of equations is important:

• They are fundamental in mathematics, used in various fields to represent relationships between quantities.
• Simplifying or solving them allows us to interpret real-world situations effectively.

Our task is to determine the values of \(x\) and \(y\) that satisfy both equations. Using methods like substitution, elimination, or Cramer's Rule helps us find these solutions efficiently. For this exercise, Cramer's Rule provides a structured approach by leveraging matrices.

Matrix Form Representation

Converting a system of equations into matrix form is a powerful technique that simplifies solving complex equations. The matrix form \[AX = B\]represents these relationships:

• \(A\) is the coefficient matrix containing the coefficients of the variables.
• \(X\) is the variable matrix, representing the unknowns of the system.
• \(B\) is the constant matrix containing the constants from the equations.

For our specific problem:\[A = \begin{bmatrix} 1 & -4 \ 3 & 8 \end{bmatrix}, \X = \begin{bmatrix} x \ y \end{bmatrix}, \B = \begin{bmatrix} -7 \ 19 \end{bmatrix}.\] By using matrix notation, we create a unified form which can be manipulated using various algebraic rules. This allows techniques like Cramer's Rule to be applied seamlessly, as matrix operations become streamlined and systematic.