Solve the following simultaneous linear equations with the help of inverse matrix method
5x + 2y - z= 12,
2x + 4y +z= 20 and 2x + y + 2z = 15

We are given the system of linear equations:

\[
5x + 2y - z = 12 \quad \text{(1)}
\]
\[
2x + 4y + z = 20 \quad \text{(2)}
\]
\[
2x + y + 2z = 15 \quad \text{(3)}
\]

We will solve this system using the inverse matrix method.

### Step 1: Write the system in matrix form

The system of equations can be written as:

\[
AX = B
\]

Where:
- \(A\) is the coefficient matrix,
- \(X\) is the column vector of variables \([x, y, z]^T\),
- \(B\) is the column vector of constants.

From the given equations, we extract:

\[
A = \begin{bmatrix} 
5 & 2 & -1 \\ 
2 & 4 & 1 \\ 
2 & 1 & 2 
\end{bmatrix}, \quad
X = \begin{bmatrix} 
x \\ 
y \\ 
z 
\end{bmatrix}, \quad
B = \begin{bmatrix} 
12 \\ 
20 \\ 
15 
\end{bmatrix}
\]

### Step 2: Find the inverse of matrix \(A\)

The solution to the system is given by:

\[
X = A^{-1}B
\]

We will now calculate the inverse of matrix \(A\). Let's compute it step by step.The solution to the system of equations is:

\[
x = 2, \quad y = 3, \quad z = 4
\]

Thus, the values of \(x\), \(y\), and \(z\) are 2, 3, and 4 respectively.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How do you determine if a matrix has an inverse?
2. What is the determinant of a matrix, and why is it important in solving systems of equations?
3. Can you use the inverse matrix method for non-square systems?
4. What other methods can be used to solve simultaneous equations?
5. How does the inverse matrix method compare to Gaussian elimination?

**Tip:** The inverse of a matrix exists only if its determinant is non-zero.

Solve the following simultaneous linear equations with the help of inverse matrix method: 5x + 2y - z= 12, 2x + 4y +z= 20 and 2x + y + 2z = 15

Learn how to solve simultaneous linear equations 5x + 2y - z = 12, 2x + 4y + z = 20, and 2x + y + 2z = 15 using the inverse matrix method. This problem demonstrates key linear algebra concepts, including matrix inversion and solving systems of equations. Suitable for high school and early undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation