Problem 58 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 2 x+3 y=-1 \\ x=\frac{y-9}{4} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{13}{7}\), \(y = -\frac{11}{7}\).

Step 1: Write the System in Standard Form

We start by writing each equation in the system in standard form (ax + by = c). The first equation is already in standard form: \(2x + 3y = -1\). For the second equation, multiply both sides by 4 to eliminate the fraction:\(x - \frac{y}{4} = -\frac{9}{4}\) becomes \(4x - y = 9\). Thus, the system becomes:\[\begin{align*}2x + 3y &= -1 \4x - y &= 9\end{align*}\]

Step 2: Set Up Matrices for the System

For Cramer's rule, identify the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\). The coefficient matrix is \[A = \begin{bmatrix} 2 & 3 \ 4 & -1 \end{bmatrix}\] the variable matrix is \[X = \begin{bmatrix} x \ y \end{bmatrix}\] and the constant matrix is \[B = \begin{bmatrix} -1 \ 9 \end{bmatrix}\]

Step 3: Calculate the Determinant of the Coefficient Matrix

The determinant of \(A\) (denoted as \(\det(A)\)) is calculated as:\[\det(A) = \begin{vmatrix} 2 & 3 \ 4 & -1 \end{vmatrix} = (2)(-1) - (3)(4) = -2 - 12 = -14\]

Step 4: Apply Cramer's Rule to Solve for x

Create matrix \(A_x\) by replacing the first column of \(A\) with matrix \(B\):\[A_x = \begin{bmatrix} -1 & 3 \ 9 & -1 \end{bmatrix}\] Calculate the determinant of \(A_x\):\[\det(A_x) = \begin{vmatrix} -1 & 3 \ 9 & -1 \end{vmatrix} = (-1)(-1) - (3)(9) = 1 - 27 = -26\] Then compute \(x\) using Cramer's Rule: \(x = \frac{\det(A_x)}{\det(A)} = \frac{-26}{-14} = \frac{13}{7}\)

Step 5: Apply Cramer's Rule to Solve for y

Create matrix \(A_y\) by replacing the second column of \(A\) with matrix \(B\):\[A_y = \begin{bmatrix} 2 & -1 \ 4 & 9 \end{bmatrix}\] Calculate the determinant of \(A_y\):\[\det(A_y) = \begin{vmatrix} 2 & -1 \ 4 & 9 \end{vmatrix} = (2)(9) - (-1)(4) = 18 + 4 = 22\] Then compute \(y\) using Cramer's Rule: \(y = \frac{\det(A_y)}{\det(A)} = \frac{22}{-14} = -\frac{11}{7}\)

Step 6: State the Solution or Conclusion

Since \(\det(A) eq 0\), the system has a unique solution. The values found using Cramer's Rule are \(x = \frac{13}{7}\) and \(y = -\frac{11}{7}\), indicating the system is consistent and independent.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. The goal is to find a common solution that satisfies all equations in the system. In the exercise you read, we have a system composed of two equations involving the variables \(x\) and \(y\):

• Equation 1: \(2x + 3y = -1\)
• Equation 2: \(x = \frac{y-9}{4}\)

To solve this system, the equations must be rearranged into standard form, \(ax + by = c\). After rearrangement, the equations become:

• \(2x + 3y = -1\)
• \(4x - y = 9\)

This allows the system to be represented in matrix form, which can be solved using various algebraic methods, including substitution, elimination, or in this case, Cramer's Rule.

Determinant

The determinant plays a crucial role in solving a system of equations using matrix methods. It is a scalar value that can be computed from the elements of a square matrix. In the context of Cramer's Rule, the determinant of the coefficient matrix helps determine whether a unique solution exists.
For the matrix \(A\) given by \(\begin{bmatrix} 2 & 3 \ 4 & -1 \end{bmatrix}\), its determinant \(\det(A)\) helped solve the system by confirming the presence of a unique solution. The equation for the determinant of a 2x2 matrix is: \[\det(A) = ad - bc\] In our example: \[\det(A) = (2)(-1) - (3)(4) = -2 - 12 = -14\] A non-zero determinant, such as \(-14\) in this case, indicates that the matrix is invertible and the system has a unique solution, which can be easily calculated using Cramer's Rule.

Consistent and Independent System

A consistent and independent system of equations has exactly one solution. This is a desirable outcome when solving systems, as it provides a single point of intersection for the equations involved. A non-zero determinant of the coefficient matrix, as computed earlier, is a key indicator of a consistent and independent system.
In this exercise, since the determinant \(\det(A)\) is \(-14\), the system:

• \(2x + 3y = -1\)
• \(4x - y = 9\)

was shown to have a unique solution, with \(x = \frac{13}{7}\) and \(y = -\frac{11}{7}\). This confirms that the lines represented by these equations intersect at only one point, proving the system is both consistent and independent. Understanding the nature of a system is fundamental to selecting the appropriate method for solution.