Math Problem Statement

Solve the following system of equations by using the inverse of the coefficient matrix.

negative 8−8xminus−yplus+66z

equals=negative 33−33

88yplus+55z

equals=1919

22xplus+33yplus+66z

equals=99

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Part 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The solution of the system is

enter your response here.

​(Simplify your answer. Type an ordered​ triple.)

B.

There are infinitely many solutions. The solutions are

xequals=enter your response here

and

yequals=enter your response here​,

where z is any real number.

​(Use integers or fractions for any numbers in the​ expression.)

C.

There is no solution.

Solution

To solve the system of equations using the inverse of the coefficient matrix, let's first represent the system in matrix form:

Given system of equations:

-8x - y + 6z &= -33 \\ 8y + 5z &= 19 \\ 2x + 3y + 6z &= 9 \end{aligned}$$ We can represent this system as $$A \cdot X = B$$, where: - $$A$$ is the coefficient matrix, - $$X$$ is the column vector of variables $$[x, y, z]$$, - $$B$$ is the column vector of constants. ### Step 1: Coefficient Matrix $$A$$ The coefficient matrix $$A$$ is: $$A = \begin{pmatrix} -8 & -1 & 6 \\ 0 & 8 & 5 \\ 2 & 3 & 6 \end{pmatrix}$$ ### Step 2: Column Vector of Variables $$X$$ The column vector of variables $$X$$ is: $$X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ ### Step 3: Column Vector of Constants $$B$$ The column vector of constants $$B$$ is: $$B = \begin{pmatrix} -33 \\ 19 \\ 9 \end{pmatrix}$$ ### Step 4: Solving using the inverse of $$A$$ We can find the solution $$X = A^{-1} \cdot B$$, but first we need to calculate $$A^{-1}$$, the inverse of the coefficient matrix. Let's compute $$A^{-1}$$ and then multiply it by $$B$$. The solution to the system of equations is: $$x = 3, \quad y = 3, \quad z = -1$$ Thus, the correct choice is **A**: The solution of the system is $$(3, 3, -1)$$. Would you like more details on how the inverse matrix was calculated or any other part of the process? Here are 5 related questions for further practice: 1. How do you calculate the inverse of a 3x3 matrix? 2. What are other methods to solve a system of linear equations besides using the inverse matrix? 3. Can all systems of equations be solved using matrix inversion? 4. What does it mean if the determinant of a matrix is zero? 5. How can you verify the solution to a system of equations? **Tip:** When solving systems of equations, if the determinant of the coefficient matrix is zero, the system may have no solution or infinitely many solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Inversion

Formulas

Matrix Equation A * X = B
Inverse of a Matrix (A^-1)
Matrix Multiplication

Theorems

Inverse Matrix Theorem
Cramer's Rule (Optional Alternative)

Suitable Grade Level

Grades 10-12