Problem 28 - Precalculus: Mathematical 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) The solution is verified. (b) Determinant is -46. (c) Unique solution exists. (d) Yes, Cramer's Rule can be used.

Step 1: Verify the given solution

To verify if \( x = -1, y = 0, z = 1 \) is a solution, substitute these values into each equation of the system. For the first equation: \( x + 2y + 6z = -1 + 2(0) + 6(1) = -1 + 0 + 6 = 5 \). This is true. For the second equation: \( -3x - 6y + 5z = -3(-1) - 6(0) + 5(1) = 3 + 0 + 5 = 8 \). This is true.For the third equation: \( 2x + 6y + 9z = 2(-1) + 6(0) + 9(1) = -2 + 0 + 9 = 7 \). This is true.Since all three equations are satisfied, \((x, y, z) = (-1, 0, 1)\) is a solution.

Step 2: Calculate the determinant of the coefficient matrix

The coefficient matrix, \( A \), is given by: \[A = \begin{bmatrix}1 & 2 & 6 \-3 & -6 & 5 \2 & 6 & 9\end{bmatrix}\]Calculate the determinant, \( \text{det}(A) \), using the formula for a 3x3 matrix: \[\text{det}(A) = 1((-6)(9) - (5)(6)) - 2((-3)(9) - (5)(2)) + 6((-3)(6) - (-6)(2))\]\[= 1(-54 - 30) - 2(-27 - 10) + 6(-18 + 12)\]\[= 1(-84) - 2(-37) + 6(-6)\]\[= -84 + 74 - 36\]\[= -46\]

Step 3: Determine uniqueness of solutions

The determinant of the coefficient matrix is \(-46\), which is non-zero. A non-zero determinant implies that the system of linear equations has a unique solution. Thus, there are no other solutions to the system beside \(x = -1, y = 0, z = 1\).

Step 4: Use of Cramer's Rule

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

Key Concepts

Determinant

The determinant is a special value calculated from a square matrix. It's a scalar that provides a multitude of insights into the matrix's properties. For our given system of linear equations, the coefficient matrix is \[A = \begin{bmatrix} 1 & 2 & 6 \ -3 & -6 & 5 \ 2 & 6 & 9 \end{bmatrix}\]The determinant of this 3x3 matrix can be found using the formula that considers the elements and minors of the matrix. Det(A) is essential because it can tell you whether the matrix is invertible or not:

• If the determinant is nonzero, the matrix is invertible and the system has a unique solution.
• If the determinant is zero, the matrix is singular, indicating possible infinite solutions or no solution.

In this case, the determinant is \(-46\), which is nonzero,ensuring a unique solution to the system.

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided the determinant of the system's coefficient matrix is non-zero. In simple terms, it allows you to find the solution by calculating determinants of matrices. Here's what you should know about Cramer's Rule:

• It only applies to systems with the same number of equations as unknowns.
• The determinant of the original matrix should not be zero for Cramer's Rule to apply.
• Each variable in the system is found by dividing the determinant of a modified version of the matrix by the determinant of the original coefficient matrix.

In our system, because we have a non-zero determinant \(-46\), Cramer's Rule can be used to solve it. Each variable (x, y, z) is determined by replacing the respective column in the original matrix with the solution vector and finding the determinant of this new matrix.

Unique Solution

The concept of a unique solution in systems of linear equations is crucial in determining the nature of the solutions. A system of equations can have:

• A unique solution
• Infinitely many solutions
• No solution

A unique solution signifies that there is exactly one solution that satisfies all equations, which occurs when the determinant of the coefficient matrix is non-zero. In our system, because the determinant \(-46\) is non-zero, this directly indicates that the solution is unique. No other solution exists other than the one found: \((x, y, z) = (-1, 0, 1)\). This uniqueness is an outcome of the invertibility of the matrix, confirming that the system of equations is consistent and independently solvable.