Problem 68 - Precalculus: Mathematics for Calculus

Question

Consider the system \(\left\\{\begin{aligned} x+2 y+6 z &=5 \\\\-3 x-6 y+5 z &=8 \\ 2 x+6 y+9 z &=7 \end{aligned}\right.\) (a) Verify that \(x=-1, y=0, z=1\) is a solution of the system. (b) Find the determinant of the coefficient matrix. (c) Without solving the system, determine whether there are any other solutions. (d) Can Cramer's Rule be used to solve this system? Why or why not?

Step-by-Step Solution

Answer

(a) Solution holds; (b) Det = -46; (c) No other solutions exist; (d) Cramer's Rule is applicable.

Step 1: Verify Given Solution

Substitute \(x = -1\), \(y = 0\), and \(z = 1\) into each equation and verify that they satisfy the system.For the first equation: \(x + 2y + 6z = -1 + 2(0) + 6(1) = 5\). This satisfies the first equation.For the second equation:\(-3x - 6y + 5z = -3(-1) - 6(0) + 5(1) = 3 + 5 = 8\). This satisfies the second equation.For the third equation:\(2x + 6y + 9z = 2(-1) + 6(0) + 9(1) = -2 + 9 = 7\). This satisfies the third equation.Thus, \(x = -1\), \(y = 0\), \(z = 1\) is a solution of the system.

Step 2: Determine the Coefficient Matrix

The coefficient matrix of the system is:\[\begin{bmatrix}1 & 2 & 6 \-3 & -6 & 5 \2 & 6 & 9\end{bmatrix}\]

Step 3: Compute the Determinant of the Coefficient Matrix

The determinant of matrix \(A\) (the coefficient matrix) is computed as:\[ \text{det}(A) = 1 \begin{vmatrix} -6 & 5 \ 6 & 9 \end{vmatrix} -2 \begin{vmatrix} -3 & 5 \ 2 & 9 \end{vmatrix} +6 \begin{vmatrix} -3 & -6 \ 2 & 6 \end{vmatrix}\]Calculate each minor determinant:1. \(-6 \times 9 - 5 \times 6 = -54 - 30 = -84\)2. \(-3 \times 9 - 5 \times 2 = -27 - 10 = -37\)3. \(-3 \times 6 - (-6) \times 2 = -18 + 12 = -6\)Therefore, \[\text{det}(A) = 1(-84) - 2(-37) + 6(-6) = -84 + 74 - 36 = -46\]

Step 4: Determine the Solvability of the System

Since the determinant of the coefficient matrix \(A\) is non-zero \((-46)\), the matrix is invertible. This implies that the system has a unique solution. Therefore, there are no other solutions except the given one.

Step 5: Evaluate Cramer's Rule Applicability

Cramer's Rule can be used to solve a system of equations if the determinant of the coefficient matrix is non-zero. Since \(\text{det}(A) = -46 eq 0\), Cramer's Rule can be used to solve this system.

Key Concepts

Determinant of a Matrix

A determinant is a special number that can be calculated from a square matrix. It is an essential concept in linear algebra, especially when dealing with systems of linear equations. For a 3x3 matrix, such as \[\begin{bmatrix}1 & 2 & 6 \-3 & -6 & 5 \2 & 6 & 9\end{bmatrix}\], the determinant provides useful information about the matrix.

• If the determinant of a matrix is non-zero, the matrix is invertible, meaning it has an inverse. This property is critical for determining if a system of linear equations has unique solutions.
• A zero determinant indicates that the matrix is singular, suggesting either no solutions or infinitely many solutions.

To compute the determinant of a 3x3 matrix like matrix \(A\), you can use the formula:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]where \((a, b, c, d, e, f, g, h, i)\) are the elements of the matrix.
This calculation helps in analyzing the solvability of a system of equations.

Cramer's Rule

Cramer's Rule is a mathematical theorem used for solving a system of linear equations with as many equations as unknowns, provided the determinant of the coefficient matrix is non-zero. It provides a straightforward way to find the value of each variable one at a time.You apply Cramer's Rule by replacing the column of the coefficients of the variable you are solving for with the constant terms from the equations. You then calculate the determinant of this new matrix (let's call it \(D_x\) for solutions involving \(x\), for instance). Finally, the value of the variable is given by:\[x = \frac{D_x}{D}\]where \(D\) is the determinant of the original coefficient matrix.

• Cramer's Rule is only applicable when the determinant of the matrix \(D\) is non-zero.
• This method is particularly useful when solving small systems of equations due to the simplicity of the calculations.

In the discussed system, since the determinant \((-46)\) is not zero, Cramer's Rule can be efficiently used to verify the solution \((x = -1, y = 0, z = 1)\).

Unique Solutions

The concept of unique solutions is central to understanding systems of linear equations. When we refer to a system having a unique solution, we mean that there is exactly one set of values for the variables that satisfies all of the equations in the system.This occurs when:

• The determinant of the coefficient matrix is non-zero, indicating that the matrix is invertible.
• The system is consistent, meaning the equations do not contradict one another.

With a non-zero determinant (such as \(-46\)), every variable has exactly one solution in the context of the system. This tells us that for the given system of equations:\[\left\{\begin{aligned} x+2y+6z &= 5 \ -3x-6y+5z &= 8 \ 2x+6y+9z &= 7 \end{aligned}\right\}\]no other solution exists except \((x = -1, y = 0, z = 1)\). This principle aids in verifying solutions reliably, ensuring that a proposed solution is not only possible but indeed the only one.