Problem 80 - Intermediate Algebra

Question

Explain how to find \(x\) when solving a system of three linear equations in \(x, y,\) and \(z\) by Cramer's rule. Use the words coefficients and constants in your explanation.

Step-by-Step Solution

Answer

Use Cramer's rule: Calculate \(\text{det}(A)\), replace the first column with constants to get \(A_x\), find \(\text{det}(A_x)\), then solve \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\).

Step 1: Understand Cramer's Rule

Cramer's rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns, when the determinant of the coefficient matrix is non-zero. For a system of three equations in variables \(x, y, z\), we seek three distinct expressions that define each variable in terms of determinants.

Step 2: Write the System of Equations

Consider a system of three linear equations: \[ \begin{align*} a_1x + b_1y + c_1z &= d_1, \ a_2x + b_2y + c_2z &= d_2, \ a_3x + b_3y + c_3z &= d_3. \end{align*} \] Here, \(a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3\) are coefficients and \(d_1, d_2, d_3\) are constants.

Step 3: Formulate the Coefficient Matrix

Create the coefficient matrix \(A\) from the coefficients of \(x, y, z\): \[ A = \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}. \] This matrix contains only the coefficients of the variables.

Step 4: Calculate the Determinant of the Coefficient Matrix

Compute the determinant of matrix \(A\), denoted as \(\text{det}(A)\). This step is crucial since Cramer's rule only applies if \(\text{det}(A) eq 0\). If the determinant is zero, the system is either dependent or inconsistent.

Step 5: Create the Modified Matrix for \(x\)

To solve for \(x\), replace the first column of \(A\) with the constants \(d_1, d_2, d_3\) to create a new matrix \(A_x\): \[ A_x = \begin{bmatrix} d_1 & b_1 & c_1 \ d_2 & b_2 & c_2 \ d_3 & b_3 & c_3 \end{bmatrix}. \] This matrix is structured by inserting the constants into the position allocated for the \(x\) coefficients.

Step 6: Calculate the Determinant for the Modified Matrix

Find the determinant of \(A_x\), noted as \(\text{det}(A_x)\). This is a critical step in solving for \(x\).

Step 7: Solve for \(x\) using Cramer's Rule

According to Cramer's Rule, \(x\) can be found by the formula: \[ x = \frac{\text{det}(A_x)}{\text{det}(A)}. \] Ensure that \(\text{det}(A)\) is not zero to validate the application of this formula.

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 simpler terms, it's like solving a puzzle where you are given equations that share unknowns, such as \(x, y,\) and \(z\). Each equation provides a piece of information that, when put together with others, helps to find a unique solution.
Linear equations can be represented in the general format of \(a_1x + b_1y + c_1z = d_1\), where \(a_1, b_1, c_1\) are the coefficients and \(d_1\) is a constant.

• The coefficients represent the number values multiplying the unknown variables.
• The constants are the numbers on the opposite side of the equation.

When solving these kinds of systems, the objective is to find values for \(x, y,\) and \(z\) that satisfy all equations simultaneously. Cramer's Rule is one of the methods used for this purpose.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a matrix, it gives insight into certain properties, such as invertibility. In the context of Cramer's Rule, the determinant is crucial. It helps determine whether a unique solution exists for the system of linear equations.
For a 3x3 matrix \(A\), which is constructed from the coefficients of the variables, the determinant is calculated using a specific formula involving its elements: \[\text{det}(A) = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)\]

• If \(\text{det}(A) = 0\), it indicates that the system might have no solution or infinitely many solutions.
• Only when \(\text{det}(A) eq 0\) can Cramer's Rule be used to find a unique solution.

Calculating the determinant involves handling the rows and columns, ensuring changes affect the sign properly, leading to accurate results.

Coefficient Matrix

The coefficient matrix is a structured arrangement of the coefficients from the system of linear equations. It effectively isolates the numerical factors that multiply the variables.
In the provided system of equations:

• \(a_1, a_2, a_3\) correspond to the coefficients of \(x\).
• \(b_1, b_2, b_3\) are for \(y\).
• \(c_1, c_2, c_3\) are for \(z\).

These are arranged in the matrix \(A\) as: \[A = \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}\]This matrix stands as a numerical representation of the core structure of the system of equations. By manipulating this matrix, especially through determinants, solutions for variables can be calculated using rules like Cramer's. This systematic approach simplifies complex problem-solving into a more manageable format, aiding in solving for unknowns.