Problem 37 - College Algebra

Question

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} 4 x+5 y-z &=-7 \\ -2 x-9 y+2 z &=8 \\ 5 y+7 z &=21 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{96}{31}, y = -\frac{210}{31}, z = \frac{332}{31} \).

Step 1: Write the system in matrix form

The given system of equations can be expressed in matrix form as \( AX = B \), where:- \( A \) is the coefficient matrix:\[A = \begin{bmatrix} 4 & 5 & -1 \ -2 & -9 & 2 \ 0 & 5 & 7 \end{bmatrix}\]- \( X \) is the column matrix of variables:\[ X = \begin{bmatrix} x \ y \ z \end{bmatrix} \]- \( B \) is the column matrix of constants:\[ B = \begin{bmatrix} -7 \ 8 \ 21 \end{bmatrix} \]

Step 2: Compute the determinant of matrix A

Using the formula for a 3x3 determinant, calculate \( \det(A) \).\[\det(A) = \begin{vmatrix} 4 & 5 & -1 \ -2 & -9 & 2 \ 0 & 5 & 7 \end{vmatrix}\]Expanding along the third column:\[\det(A) = (-1) \left( (-2)(7) - (-9)(5) \right) - (5) \left( (-2)(0) - (-9)(0) \right) + (4) \left( 5(2) - (-9)(0) \right)\]Simplifying:\[\det(A) = -1(-14 + 45) + 0 + 20 = -31\]

Step 3: Compute determinant of matrix A_x

Replace the first column of \( A \) with \( B \) to find \( A_x \):\[A_x = \begin{bmatrix} -7 & 5 & -1 \ 8 & -9 & 2 \ 21 & 5 & 7 \end{bmatrix}\]Calculate \( \det(A_x) \):\[\det(A_x) = \begin{vmatrix} -7 & 5 & -1 \ 8 & -9 & 2 \ 21 & 5 & 7 \end{vmatrix}\]Expand along the first column:\[\det(A_x) = -7 \left( (-9)(7) - 2(5) \right) - 8 \left( 5(7) - 2(-1) \right) + 21 \left( 5(-9) - (-1)(-9) \right)\]Simplifying:\[\det(A_x) = -7(-63 - 10) - 8(35 + 2) + 21(-45 - 1) = -96\]

Step 4: Compute determinant of matrix A_y

Replace the second column of \( A \) with \( B \) to find \( A_y \):\[A_y = \begin{bmatrix} 4 & -7 & -1 \ -2 & 8 & 2 \ 0 & 21 & 7 \end{bmatrix}\]Calculate \( \det(A_y) \):\[\det(A_y) = \begin{vmatrix} 4 & -7 & -1 \ -2 & 8 & 2 \ 0 & 21 & 7 \end{vmatrix}\]Expand along the first row:\[\det(A_y) = 4 \left( 8(7) - 2(21) \right) - (-7) \left( (-2)(7) - 2(0) \right) - (-1) \left( (-2)(21) - 8(0) \right)\]Simplifying:\[\det(A_y) = 4(56 - 42) + 7(14) + 42 = 210\]

Step 5: Compute determinant of matrix A_z

Replace the third column of \( A \) with \( B \) to find \( A_z \):\[A_z = \begin{bmatrix} 4 & 5 & -7 \ -2 & -9 & 8 \ 0 & 5 & 21 \end{bmatrix}\]Calculate \( \det(A_z) \):\[\det(A_z) = \begin{vmatrix} 4 & 5 & -7 \ -2 & -9 & 8 \ 0 & 5 & 21 \end{vmatrix}\]Expand along the third column:\[\det(A_z) = -7 \left( (-9)(21) - 8(5) \right) + 8 \left( 5(2) - 5(0) \right) + 21 \left( 5(-2) - 5(0) \right)\]Simplifying:\[\det(A_z) = -7(-189 - 40) + 80 + 21(-10) = -332\]

Step 6: Solve for x, y, and z using Cramer's Rule

Using Cramer's Rule, the solutions for \( x \), \( y \), and \( z \) are given by\[ x = \frac{\det(A_x)}{\det(A)} = \frac{-96}{-31} = \frac{96}{31} \]\[ y = \frac{\det(A_y)}{\det(A)} = \frac{210}{-31} = -\frac{210}{31} \]\[ z = \frac{\det(A_z)}{\det(A)} = \frac{-332}{-31} = \frac{332}{31} \]

Step 7: Conclusion of the solution

The solution to the system of equations using Cramer's Rule is \( x = \frac{96}{31} \), \( y = -\frac{210}{31} \), \( z = \frac{332}{31} \). These are the values of the variables that satisfy all three equations in the system.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In our scenario, we have three equations with three variables: \( x \), \( y \), and \( z \). The equations are presented as follows:

• 4x + 5y - z = -7
• -2x - 9y + 2z = 8
• 5y + 7z = 21

Each equation represents a plane in three-dimensional space, and the solution to the system is the point where all these planes intersect. Depending on the number of variables and equations, systems can have one solution, no solution, or infinitely many solutions.
In our case, Cramer's Rule is a method used to find the unique solution for a system of linear equations that have as many equations as unknowns, provided the determinant is non-zero.

Determinants

Determinants are a special number that can be calculated from a square matrix. They offer a wealth of information about the matrix and the system it represents. For instance, a non-zero determinant of the coefficient matrix signals that the system of linear equations has a unique solution.
In this particular exercise, we calculate determinants to use in Cramer's Rule which requires computing the determinant of the original matrix (\( A \)) and modified matrices (\( A_x \), \( A_y \), \( A_z \)) to find the solutions for the variables \( x \), \( y \), and \( z \). Here is how you can view the role of each determinant:

• \( \det(A) \) is the determinant of the coefficient matrix.
• \( \det(A_x) \), \( \det(A_y) \), and \( \det(A_z) \) are determinants of matrices derived from A by replacing columns with the constant matrix. These determine the values of \( x \), \( y \), and \( z \) respectively.

Understanding determinants helps us confidently apply and solve systems of equations using rules and properties like those found in Cramer's Rule.

Matrix Form

Expressing a system of linear equations in matrix form is a compact and elegant way to represent and solve complex systems. To use matrix operations, we first rearrange our given equations into a matrix format. For our problem, the matrices have the following form:

• Coefficient matrix (\( A \)) which holds the coefficients of variables.
• Variable matrix (\( X \)) which consists of the variables we need to solve for.
• Constant matrix (\( B \)) which includes the constants from the right side of each equation.

The system \( AX = B \) succinctly represents the given collection of linear equations. This format enables us to apply various linear algebra techniques such as multiplication, finding determinants, and utilizing Cramer's Rule to efficiently solve the equations. Once the system is in matrix form, the problem becomes more manageable and allows for systematic procedures, like calculating determinants and applying Cramer's Rule, to find precise solutions efficiently.