Problem 36 - College Algebra

Question

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{l}3 x+2 z=4 \\\5 x-y=-4 \\\4 y+3 z=22\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = \frac{28}{31}\), \(y = 4\), \(z = 12\).

Step 1: Create the Matrices

First, create four 3x3 matrices: The coefficient matrix \(D = \begin{bmatrix}3 & 0 & 2 \5 & -1 & 0 \0 & 4 & 3 \end{bmatrix}\); The X-matrix \(D_x = \begin{bmatrix}4 & 0 & 2 \-4 & -1 & 0 \22 & 4 & 3 \end{bmatrix}\); The Y-matrix \(D_y = \begin{bmatrix}3 & 4 & 2 \5 & -4 & 0 \0 & 22 & 3 \end{bmatrix}\); The Z-matrix \(D_z = \begin{bmatrix}3 & 0 & 4 \5 & -1 & -4 \0 & 4 & 22 \end{bmatrix}\)

Step 2: Compute the Determinants

Compute the determinant for each matrix using the determinant rule for 3x3 matrices. \(|D| = 3((-1*3) - (0)) - 0(15 - 0) + 2(20 - 0) = -9 + 0 + 40 = 31 \) \(|D_x| = 4((-1*3) - (0)) - 0(15 - 0) + 2(20 - 0) = -12 + 0 + 40 = 28 \) \(|D_y| = 3((-4*3) - (0)) - 4(15 - 0) + 2(22*5 - 0) = -36 -60 + 220 = 124 \) \(|D_z| = 3((-1*22) - (0)) - 0(15 - 0) + 4(22*5 - 0) = -66 + 0 + 440 = 374 \)

Step 3: Solve for x, y and z

Solve for x, y, and z by dividing \(|D_x|\), \(|D_y|\) and \(|D_z|\) by \(|D|\). \(x = \frac{|D_x|}{|D|} = \frac{28}{31}\) \(y = \frac{|D_y|}{|D|} = \frac{124}{31} = 4\) \(z = \frac{|D_z|}{|D|} = \frac{374}{31} = 12\)

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share a common set of variables. Solving such a system means finding the values of the variables that satisfy all equations simultaneously.

For example, the given system of equations from this exercise describes a three-dimensional space where each equation represents a plane. The solution to the system is the point where these planes intersect.

The system:

• 3x + 2z = 4
• 5x - y = -4
• 4y + 3z = 22

To find these intersection points conveniently, Cramer's Rule can be applied, which works effectively for systems that have the same number of equations as variables and where the determinant of the coefficient matrix is non-zero. This rule provides a direct method of finding the solution using determinants, which is often more efficient than other methods such as substitution or elimination, particularly for systems where the number of equations is small.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. The determinant of a matrix provides important information about the matrix, such as whether it is invertible or not.

For a 3x3 matrix, the determinant can be found using a specific formula that involves the multiplication and subtraction of certain matrix elements arranged in a particular pattern. In the context of Cramer's Rule, determinants are crucial because they are used to calculate the values of the variables.

In the exercise, determinant calculations are as follows for the coefficient matrix and each modified matrix:

• \(|D| = -9 + 40 = 31\)
• \(|D_x| = -12 + 40 = 28\)
• \(|D_y| = -36 - 60 + 220 = 124\)
• \(|D_z| = -66 + 440 = 374\)

The non-zero determinant of the coefficient matrix \(D\) means that the system of equations has a unique solution. This is a fundamental prerequisite for applying Cramer's Rule.

Matrices

A matrix is a rectangular array of numbers, often used to represent coefficients in a system of linear equations. Each row of the matrix corresponds to an equation, and each column corresponds to a variable.

In the context of solving systems using Cramer's Rule, several matrices are constructed:

• The coefficient matrix, D, which comprises the coefficients of the variables from the system.
• Matrices D_x, D_y, and D_z, which are created by replacing the respective variable's column in D with the constants from the right side of the equations.

Through the exercise, you were introduced to forming these matrices as a first step in implementing Cramer's Rule:

• D contains the coefficients from the original system, forming the 'skeleton' for other matrices.
• D_x replaces the x-column with the constant terms to solve for x.
• D_y replaces the y-column for solving y, and similarly, D_z for z.

Once we have the determinants of these matrices, Cramer's Rule utilizes them to find the solution to the system, reflecting the close relationship between algebraic equations and matrix operations.