Problem 22 - Elementary and Intermediate Algebra

Question

Solve using Cramer's rule. $$ \begin{aligned} x-y+2 z &=-3 \\ x+2 y+3 z &=4 \\ 2 x+y+z &=-3 \end{aligned} $$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{5}{6} \), \( y = 2 \), \( z = 1 \).

Step 1: Write the system of linear equations in matrix form

Express the system of equations as a matrix equation:\[\begin{pmatrix} 1 & -1 & 2 \ 1 & 2 & 3 \ 2 & 1 & 1 \end{pmatrix}\begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} -3 \ 4 \ -3 \end{pmatrix}\]

Step 2: Find the determinant of the coefficient matrix

Calculate the determinant of the coefficient matrix:\[\Delta = \begin{vmatrix} 1 & -1 & 2 \ 1 & 2 & 3 \ 2 & 1 & 1 \end{vmatrix}\].Use the formula for determinants:\[\Delta = 1(2 \cdot 1 - 3 \cdot 1) - (-1)(1 \cdot 1 - 3 \cdot 2) + 2(1 \cdot 1 - 2 \cdot 2) = 1(-1) - (-1)(-5) + 2(-3) = -1 - 5 - 6 = -12\]

Step 3: Find the determinant of the matrix for x

Replace the first column of the coefficient matrix with the constants vector and find the determinant:\[\Delta_x = \begin{vmatrix} -3 & -1 & 2 \ 4 & 2 & 3 \ -3 & 1 & 1 \end{vmatrix}\].Calculate the determinant:\[\Delta_x = (-3)(2 \cdot 1 - 1 \cdot 3) - (-1)(4 \cdot 1 - (-3) \cdot 3) + 2(4 \cdot 1 - 2 \cdot (-3)) = -3(-1) - (-1)(13) + 2(10) = 3 - 13 + 20 = 10\]

Step 4: Find the determinant of the matrix for y

Replace the second column of the coefficient matrix with the constants vector and find the determinant:\[\Delta_y = \begin{vmatrix} 1 & -3 & 2 \ 1 & 4 & 3 \ 2 & -3 & 1 \end{vmatrix}\].Calculate the determinant:\[\Delta_y = 1(4 \cdot 1 - 3 \cdot (-3)) - (-3)(1 \cdot 1 - 3 \cdot 2) + 2(1 \cdot (-3) - 4 \cdot 2) = 1(13) - (-3)(-5) + 2(-11) = 13 - 15 - 22 = -24\]

Step 5: Find the determinant of the matrix for z

Replace the third column of the coefficient matrix with the constants vector and find the determinant:\[\Delta_z = \begin{vmatrix} 1 & -1 & -3 \ 1 & 2 & 4 \ 2 & 1 & -3 \end{vmatrix}\].Calculate the determinant:\[\Delta_z = 1(2 \cdot (-3) - 4 \cdot 1) - (-1)(1 \cdot (-3) - 4 \cdot 2) + (-3)(1 \cdot 1 - 2 \cdot 2) = 1(-10) - (-1)(-11) + (-3)(-3) = -10 - 11 + 9 = -12\]

Step 6: Solve for the variables using Cramer's Rule

Use Cramer's Rule to find values of x, y, and z:\[x = \frac{\Delta_x}{\Delta} = \frac{10}{-12} = -\frac{5}{6} \ y = \frac{\Delta_y}{\Delta} = \frac{-24}{-12} = 2 \ z = \frac{\Delta_z}{\Delta} = \frac{-12}{-12} = 1\]

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The equations are typically written like this:

• Equation 1: \( a_1x + b_1y + c_1z = d_1 \)
• Equation 2: \( a_2x + b_2y + c_2z = d_2 \)
• Equation 3: \( a_3x + b_3y + c_3z = d_3 \)

Here, the goal is to find the values of the variables \(x\), \(y\), and \(z\) that simultaneously satisfy all the given equations. Solving these systems can be done using various methods, one of which is Cramer's rule. This approach is particularly efficient for smaller systems.

Matrix Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides essential properties about the matrix. Let's consider a 3x3 matrix, denoted by \( A \):\[ A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \]The determinant of matrix \( A \), denoted as \( \Delta \) or \( \text{det}(A) \), is calculated as follows:\[ \Delta = a(ei - fh) - b(di - fg) + c(dh - eg) \]Here, the calculation involves the sum and difference of products of the elements of the matrix, reflecting its geometric and algebraic properties. The determinant value can inform you if a matrix is singular (i.e., non-invertible) when it equals zero, or invertible when it doesn't. Matrix determinants are particularly useful in Cramer's rule for solving systems of linear equations.

Solving Linear Equations

To solve a system of linear equations using Cramer's rule, you need to follow these steps:

• Write the system of equations in matrix form.
• Find the determinant of the coefficient matrix (\( \Delta \)).
• Replace each column of the coefficient matrix by the constants vector to find the determinants for each variable (\( \Delta_x, \Delta_y, \Delta_z \)).
• Compute each variable using the ratios of these determinants: \( x = \frac{\Delta_x}{\Delta}, y = \frac{\Delta_y}{\Delta}, z = \frac{\Delta_z}{\Delta} \).

For instance, consider solving the system of equations given in the problem:\[ \begin{aligned} x - y + 2z &= -3 \ x + 2y + 3z &= 4 \ 2x + y + z &= -3 \end{aligned} \]Following the steps, you first write the coefficient matrix and constants vector, find the determinants, and finally compute the values of \(x\), \(y\), and \(z\) using the calculated determinants. This systematic approach ensures that the solution is clear and accurate.