Math Problem Statement

Solve the following system of equation by method of inversion x + 4y + z = 10 , 2x - y + z = 1 , yy + z = 8

Solution

To solve the system of equations using the matrix inversion method, we first rewrite the system of equations in matrix form.

Given system:

x + 4y + z &= 10 \\ 2x - y + z &= 1 \\ y + z &= 8 \end{aligned}$$ ### Step 1: Write the system in matrix form We can write the system as $$A \mathbf{x} = \mathbf{b}$$, where: $$A = \begin{pmatrix} 1 & 4 & 1 \\ 2 & -1 & 1 \\ 0 & 1 & 1 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 10 \\ 1 \\ 8 \end{pmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ To solve for $$\mathbf{x}$$, we need to find the inverse of $$A$$, i.e., $$A^{-1}$$. Once we have $$A^{-1}$$, we can compute: $$\mathbf{x} = A^{-1} \mathbf{b}$$ Let me calculate the inverse of $$A$$ and then solve the system.The solution to the system of equations is: $$x = -2.125, \quad y = 1.375, \quad z = 6.625$$ This means the values of $$x$$, $$y$$, and $$z$$ that satisfy the system of equations are approximately: $$x = -2.125, \quad y = 1.375, \quad z = 6.625$$ Would you like further details or clarification? Here are 5 related questions you might consider: 1. How can you verify the solution by substituting values back into the original equations? 2. What are the conditions under which matrix inversion is possible? 3. Can all systems of equations be solved using the matrix inversion method? 4. What happens if matrix $$A$$ is not invertible (singular)? 5. How does the determinant of a matrix relate to the existence of an inverse? **Tip:** When solving systems of equations, checking the determinant of the coefficient matrix helps ensure that the matrix inversion method is valid. If the determinant is zero, the matrix is singular and not invertible.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Inversion

Formulas

Matrix form: A * x = b
Inverse of matrix A: A^-1
Solution of the system: x = A^-1 * b

Theorems

Matrix Inversion Theorem
Determinant and Inversibility

Suitable Grade Level

Grades 10-12

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