Problem 53 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. $$ \left\\{\begin{array}{l} 3 x+2 y-z=-8 \\ 2 x-y+7 z=10 \\ 2 x+2 y-3 z=-10 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 41, y = -22, z = 12 \).

Step 1: Write the System in Matrix Form

First, represent the system of equations as a matrix equation of the form \( AX = B \). Thus, the coefficient matrix \( A \), the variable matrix \( X \), and the constant matrix \( B \) are:\[A = \begin{bmatrix}3 & 2 & -1 \2 & -1 & 7 \2 & 2 & -3\end{bmatrix}, X = \begin{bmatrix} x \ y \ z\end{bmatrix}, B = \begin{bmatrix}-8 \10 \-10\end{bmatrix}.\]

Step 2: Calculate the Determinant of Matrix A

Calculate the determinant of the coefficient matrix \( A \) using the formula for a 3x3 matrix determinant:\[ \text{det}(A) = 3((-1)(-3) - 2\cdot 7) - 2(2(-3) - 2\cdot 7) + (-1)(2\cdot 2 - 2\cdot (-1)).\]Calculate each part:- \(3((-1)(-3) - 2\cdot 7) = 3(3 - 14) = 3(-11) = -33\)- \(2(2(-3) - 2\cdot 7) = 2(-6 - 14) = 2(-20) = -40\)- \((-1)(4 + 2) = -1 \cdot 6 = -6\)Sum them: \(-33 + 40 - 6 = 1\). Thus, \( \text{det}(A) = 1 \).

Step 3: Construct Matrices A_x, A_y, A_z

Construct matrices \( A_x, A_y, A_z \) by replacing the respective columns of \( A \) with \( B \).Matrix \( A_x \) (replace the first column of \( A \) with \( B \)):\[\begin{bmatrix}-8 & 2 & -1 \10 & -1 & 7 \-10 & 2 & -3\end{bmatrix}\]Matrix \( A_y \) (replace the second column of \( A \) with \( B \)):\[\begin{bmatrix}3 & -8 & -1 \2 & 10 & 7 \2 & -10 & -3\end{bmatrix}\]Matrix \( A_z \) (replace the third column of \( A \) with \( B \)):\[\begin{bmatrix}3 & 2 & -8 \2 & -1 & 10 \2 & 2 & -10\end{bmatrix}\]

Step 4: Calculate Determinants of A_x, A_y, A_z

Calculate each of the determinants:- Determinant of \( A_x \):\[(-8)(-1)(-3) + 2(7)(-10) + (-1)(10)(2) - ((-1)(-10)(2) + (-8)(7)(2) + (-3)10(2))\]Calculation steps lead to \( \text{det}(A_x) = 41 \).- Determinant of \( A_y \):\[3(10)(-3) + (-8)7(2) + (-1)2(-10) - ((-1)10(2) + (-3)7(2) + (-8)2(-3))\]Calculation steps lead to \( \text{det}(A_y) = -22 \).- Determinant of \( A_z \):\[3((-1)(-10) - 2\cdot 2) - 2(2(-10) - 2\cdot 2) + (-8)(2(-1) - (-10)2)\]Calculation steps lead to \( \text{det}(A_z) = 12 \).

Step 5: Solve for Variables Using Cramer's Rule

Using Cramer's Rule, solve for \( x, y, ext{ and } z \) by dividing the determinants of \( A_x, A_y, \text{ and } A_z \) by \( \text{det}(A) \):\[x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{41}{1} = 41\]\[y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-22}{1} = -22\]\[z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{12}{1} = 12\]Thus, the solution to the system is \( x = 41, y = -22, z = 12 \).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In our case, we have three equations involving three variables: \(x\), \(y\), and \(z\). These equations represent linear relationships. Solving such a system means finding values of the variables that satisfy all the equations simultaneously. This system is represented by:

• 3x + 2y - z = -8
• 2x - y + 7z = 10
• 2x + 2y - 3z = -10

Each equation forms a plane in three-dimensional space. The solution to the system is the point where all these planes intersect. By using methods like Cramer's Rule or others such as substitution or elimination, we can find the point of intersection, which are the values of \(x\), \(y\), and \(z\) that solve all three equations.

Determinant Calculation

Calculating the determinant is a key step when solving systems of equations using matrices. The determinant helps in understanding whether a unique solution exists for a system. For a 3x3 matrix, the determinant is a specific arithmetic combination of its elements. The formula for calculating the determinant of matrix \(A\) is:\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]where \(a, b, c, d, e, f, g, h, i\) are the elements of the matrix \(A\).

In our solution, the determinant of matrix \(A\) was found to be 1. This determinant shows that the matrix is invertible and thus the system has a unique solution. In cases where the determinant is zero, the system may be either dependent (infinitely many solutions) or inconsistent (no solutions).

Matrix Algebra

Matrix algebra is the mathematical tool used to represent and solve linear equations in a compact form. In our exercise, the system of equations is expressed as a matrix equation:\[ AX = B \]where \(A\) is the coefficient matrix, \(X\) is the variable matrix containing \(x\), \(y\), and \(z\), and \(B\) is the constant matrix. This representation allows us to apply operations such as determinant calculation and matrix multiplication to solve the system efficiently.

A handy method to solve these systems is Cramer's Rule, which uses determinants to find the values of variables. Choosing matrix representations simplifies complex algebraic manipulations, making it easier to identify matrix types, perform operations, and ultimately solve systems of equations.