Problem 26 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rr} 8 x-11 y= & 3 \\ -x+4 y= & -3 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \(x = -1\) and \(y = -1\).

Step 1: Write the given system in augmented matrix form

The system of equations provided is:\[\begin{align*}8x - 11y &= 3 \,\-x + 4y &= -3 \end{align*}\]Correspondingly, we can express this system in matrix form as:\[\begin{bmatrix}8 & -11 \-1 & 4 \end{bmatrix} \cdot \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 3 \ -3 \end{bmatrix}\]

Step 2: Calculate the determinant of the coefficient matrix

Let's find the determinant of the coefficient matrix \[A = \begin{bmatrix}8 & -11 \-1 & 4 \end{bmatrix}\]The determinant \(\text{det}(A)\) is calculated as follows:\[\text{det}(A) = (8)(4) - (-11)(-1) = 32 - 11 = 21\]

Step 3: Calculate the determinant of matrix \(A_x\)

To find the matrix \(A_x\), replace the first column of \(A\) with the constants from the right side of the equations:\[A_x = \begin{bmatrix}3 & -11 \-3 & 4 \end{bmatrix}\]Calculate \(\text{det}(A_x)\):\[\text{det}(A_x) = (3)(4) - (-11)(-3) = 12 - 33 = -21\]

Step 4: Calculate the determinant of matrix \(A_y\)

For matrix \(A_y\), replace the second column of \(A\) with the constants from the right side:\[A_y = \begin{bmatrix}8 & 3 \-1 & -3 \end{bmatrix}\]Calculate \(\text{det}(A_y)\):\[\text{det}(A_y) = (8)(-3) - (3)(-1) = -24 + 3 = -21\]

Step 5: Apply Cramer's Rule to find the values of \(x\) and \(y\)

Using Cramer's Rule, which states that \(x\) and \(y\) can be found using:\[x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)}\]Substitute the given determinants:\[x = \frac{-21}{21} = -1, \quad y = \frac{-21}{21} = -1\]Thus, the solution is \(x = -1\) and \(y = -1\).

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix, providing useful properties about the matrix. In the context of a system of linear equations, the determinant helps us understand whether a unique solution exists. If the determinant is zero, the system might be dependent or have no solutions. A non-zero determinant, however, signifies that a unique solution is available.

• For a 2x2 matrix \(\begin{bmatrix}a & b \ c & d\end{bmatrix}\), the determinant \(\text{det}(A)\) can be calculated using the formula: \[\text{det}(A) = ad - bc\]
• This value represents an area-related concept geometrically, but it plays a crucial role in algebra for solving equations using methods like Cramer's Rule.

Augmented Matrix

An augmented matrix is a way of representing a system of linear equations in one compact format. It combines the coefficient matrix (linear terms) and the constants from the equations to form a larger matrix.

• This representation is convenient for performing operations needed to solve the system, such as row reduction or using Cramer's Rule.
• For the system \(8x - 11y = 3\) and \(-x + 4y = -3\), the augmented matrix is \(\begin{bmatrix}8 & -11 & | & 3 \ -1 & 4 & | & -3\end{bmatrix}\).
• The vertical bar separates the coefficients and constants for clarity, but it's mostly a visual aid as the matrix operations consider the entire construct.

System of Equations

A system of equations is a collection of two or more equations with a common set of unknowns. Solving this system means finding the values of variables that satisfy each equation simultaneously.

• In linear systems, solutions can be found using various methods such as graphing, substitution, elimination, or matrix approaches like Cramer's Rule.
• Linear systems are common in many areas of science and engineering where relationships between variables must be determined.
• This method ensures all equations are simultaneously true, providing insights into how variables interact with each other.

Determinant Calculation

Calculating determinants is an important step in methods like Cramer's Rule for solving systems of equations. For a 2x2 matrix, this involves multiplying diagonals and subtracting them.

• The process starts by setting up the coefficient matrix \(\begin{bmatrix}a & b \ c & d\end{bmatrix}\) and calculating the determinant: \(ad - bc\).
• Different determinant calculations help find specific variables by incorporating the constants—swap the respective columns of the determinant matrix.
• In Cramer's Rule, once the determinants are calculated, variables are determined by dividing the respective determinant's value by the original determinant of the coefficient matrix.

This means if the determinant of the coefficient matrix is known, then calculating \(\text{det}(A_x)\) and \(\text{det}(A_y)\) allows for direct computation of the solutions.