Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Use Cramer's rule to solve the system for \(x\). $$ \left\\{\begin{array}{l} a x+b y+c z=d \\ e x+f z=g \\ h x+i y=j \end{array}\right. $$

Step-by-Step Solution

Answer

Use Cramer's Rule, compute determinants, and solve for \( x \).

Step 1: Write Down the Coefficient Matrix

First, establish the coefficient matrix \( A \) from the linear equations system:\[A = \begin{bmatrix} a & b & c \ e & 0 & f \ h & i & 0 \end{bmatrix}\]This matrix represents the coefficients of \( x \), \( y \), and \( z \) from the given equations.

Step 2: Calculate the Determinant of the Coefficient Matrix

Determine the determinant \( \det(A) \) of the coefficient matrix \( A \):\[\det(A) = \begin{vmatrix} a & b & c \ e & 0 & f \ h & i & 0 \end{vmatrix}\]This requires calculating the determinant using the rule for 3x3 matrices, which expands into:\[a(0f - fi) - b(ef - fh) + c(ei - 0h)\].

Step 3: Determine the Modified Matrix for \( x \)

Replace the first column of \( A \) (the \( x \) coefficients) with the constants from the right-hand side of the equations, forming matrix \( A_x \):\[A_x = \begin{bmatrix} d & b & c \ g & 0 & f \ j & i & 0 \end{bmatrix}\]

Step 4: Calculate the Determinant of the Modified Matrix

Compute the determinant \( \det(A_x) \) of the modified matrix \( A_x \):\[\det(A_x) = \begin{vmatrix} d & b & c \ g & 0 & f \ j & i & 0 \end{vmatrix}\]Similar to previous determinant calculation, use the same expansion method:\[d(0f - fi) - b(gf - fj) + c(gi - 0j)\].

Step 5: Apply Cramer's Rule to Find \( x \)

Using Cramer's Rule, solve for \( x \):\[x = \frac{\det(A_x)}{\det(A)}\]Ensure that \( \det(A) eq 0 \); otherwise, the system does not have a unique solution.

Key Concepts

Coefficient Matrix

The coefficient matrix is essential in solving a system of linear equations, like those given in the original exercise. A coefficient matrix, such as matrix \( A \) in our frame of reference, is a matrix used to hold all the coefficients corresponding to variables in each linear equation. For this exercise, the coefficient matrix is given by: \[A = \begin{bmatrix} a & b & c \ e & 0 & f \ h & i & 0 \end{bmatrix}\]This arrangement precisely holds the coefficients from equations for \( x \), \( y \), and \( z \). Each row corresponds to a single equation, while each column corresponds to one particular variable.

• First column holds coefficients for \( x \)
• Second column holds coefficients for \( y \)
• Third column holds coefficients for \( z \)

Identifying and understanding the coefficient matrix is the first step in employing Cramer's Rule.

Determinant

The determinant of the coefficient matrix plays a crucial role in Cramer's Rule. The determinant helps determine if a unique solution exists. For a 3x3 matrix like \( A \), the determinant is calculated as follows: \[\det(A) = \begin{vmatrix} a & b & c \ e & 0 & f \ h & i & 0 \end{vmatrix}\]To find the determinant, notice the multi-step expansion along the first row:

• Multiply \( a \) by the determinant of the 2x2 minor formed by removing the row and column containing \( a \)
• Subtract \( b \) times the determinant of its corresponding minor
• Add \( c \) times the determinant of its corresponding minor

This process will produce a scalar value. If this determinant is zero, an alternate method must be used because the unique solution cannot exist using Cramer's Rule.

Linear Equations System

A linear equations system consists of a set of linear equations that are solved simultaneously to find a common solution. These equations involve one or more variables, with each term being a constant or a product of a constant with a single variable. In our exercise, we deal with:

• \( ax + by + cz = d \)
• \( ex + fz = g \)
• \( hx + iy = j \)

These represent a linear equations system involving three unknowns: \( x \), \( y \), and \( z \). Solving it involves finding the values of these variables that satisfy all equation constraints. When represented in matrix form using the coefficient matrix, simpler methods like Cramer's Rule become applicable.

Unique Solution

In the context of linear equations, a unique solution implies that there is only one set of values for the unknown variables that satisfies all of the equations simultaneously.

• The determinant \( \det(A) \) is a non-zero value, confirming that a unique solution exists.
• Each equation contributes necessary constraints to pinpoint a distinct solution.

Using Cramer's Rule, when a unique solution exists, we replace one variable's coefficients with the constants to form a modified matrix, such as \( A_x \) for \( x \). This results in:\[A_x = \begin{bmatrix} d & b & c \ g & 0 & f \ j & i & 0 \end{bmatrix}\]For \( x \), solve using Cramer's Rule:\[x = \frac{\det(A_x)}{\det(A)}\]This formula only works when the original matrix's determinant is non-zero, ensuring a unique solution.