Problem 19 - Precalculus

Question

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} 3 x+5 y=16 \\ y-x=0 \end{array}$$

Step-by-Step Solution

Answer

The solution is \( x = 2 \) and \( y = 2 \).

Step 1: Write the System in Matrix Form

Write the system of equations as follows: \[\begin{align*}3x + 5y &= 16 \y - x &= 0\end{align*}\] The corresponding matrix equation is \( A \mathbf{x} = \mathbf{b} \), where \( A = \begin{pmatrix} 3 & 5 \ -1 & 1 \end{pmatrix}, \, \mathbf{x} = \begin{pmatrix} x \ y \end{pmatrix}, \, \mathbf{b} = \begin{pmatrix} 16 \ 0 \end{pmatrix} \).

Step 2: Calculate the Determinant of the Coefficient Matrix

Calculate the determinant of matrix \( A \):\[\det(A) = \begin{vmatrix} 3 & 5 \ -1 & 1 \end{vmatrix} = (3)(1) - (5)(-1) = 3 + 5 = 8\] Since the determinant is non-zero (\( \det(A) = 8 \)), Cramer's rule can be applied.

Step 3: Find the Determinant for \( x \)

Replace the first column of \( A \) with \( \mathbf{b} \) to form \( A_x \) and calculate \( \det(A_x) \):\[A_x = \begin{pmatrix} 16 & 5 \ 0 & 1 \end{pmatrix}\]\[\det(A_x) = \begin{vmatrix} 16 & 5 \ 0 & 1 \end{vmatrix} = (16)(1) - (5)(0) = 16\]

Step 4: Find the Determinant for \( y \)

Replace the second column of \( A \) with \( \mathbf{b} \) to form \( A_y \) and calculate \( \det(A_y) \):\[A_y = \begin{pmatrix} 3 & 16 \ -1 & 0 \end{pmatrix}\]\[\det(A_y) = \begin{vmatrix} 3 & 16 \ -1 & 0 \end{vmatrix} = (3)(0) - (16)(-1) = 0 + 16 = 16\]

Step 5: Apply Cramer's Rule

Use Cramer's rule to find \( x \) and \( y \):\[x = \frac{\det(A_x)}{\det(A)} = \frac{16}{8} = 2\]\[y = \frac{\det(A_y)}{\det(A)} = \frac{16}{8} = 2\]Therefore, the solution of the system is \( x = 2 \), \( y = 2 \).

Key Concepts

System of Equations

A system of equations refers to a collection of two or more equations that have common variables. In the context of algebra, the main goal of solving such a system is to find the values of these variables that satisfy all the given equations simultaneously. For instance, in the system given in the exercise, we have two equations:

• \( 3x + 5y = 16 \)
• \( y - x = 0 \)

Both equations share the variables \( x \) and \( y \). Solving a system of equations may involve methods such as substitution, elimination, or matrix-based approaches like Cramer's Rule. The choice of method can depend on the characteristics of the system—such as the number of equations and variables—and sometimes on personal preference.
However, it's important to first determine whether a solution is achievable. This step involves checking the determinant when using matrix methods, ensuring the equations are neither contradictory nor redundant.

Determinant

A determinant is a special number that can be calculated from a matrix, which provides important information about the matrix's properties. Determinants are central in determining whether a system of linear equations can be solved using matrix methods, such as Cramer's Rule. For a matrix with two rows and two columns, like the one in the exercise, the determinant is calculated using this formula:

• For a matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is \( \ \det(A) = ad - bc \).

In our solution, where \( A = \begin{pmatrix} 3 & 5 \ -1 & 1 \end{pmatrix} \), the determinant is: \[ \det(A) = (3)(1) - (5)(-1) = 3 + 5 = 8. \]Because the determinant is non-zero, we are assured that the system of equations described by this matrix can be solved uniquely. This non-zero property indicates that the matrix is invertible, making Cramer's Rule applicable for finding the solution to the system.

Matrix Form

Writing a system of equations in matrix form is a powerful tool for solving them, especially when dealing with multiple linear equations. The process involves constructing a matrix equation of the form:

• \( A \mathbf{x} = \mathbf{b} \)

where \( A \) is the coefficient matrix containing the coefficients of the variables, \( \mathbf{x} \) is the column matrix of variables, and \( \mathbf{b} \) is the column matrix of constants from the equations' right-hand sides. For the example in the exercise:

• The coefficient matrix \( A = \begin{pmatrix} 3 & 5 \ -1 & 1 \end{pmatrix} \)
• Variable matrix \( \mathbf{x} = \begin{pmatrix} x \ y \end{pmatrix} \)
• Constant matrix \( \mathbf{b} = \begin{pmatrix} 16 \ 0 \end{pmatrix} \)

By rewriting systems algebraically into matrix form, you set up the problem to be solved using linear algebra techniques like matrix inverses or Cramer's Rule. This process simplifies handling larger systems and provides a structured approach to finding solutions efficiently.