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Solve the following system of equations:

[tex]\[ \left\{ \begin{array}{l}
5x - 2y + z = 24 \\
2x + 5y + 2z = -14 \\
x - 4y + 3z = 26
\end{array} \right. \][/tex]

Sagot :

Certainly! Let's solve the given system of linear equations step by step. The system is as follows:

[tex]\[ \begin{cases} 5x - 2y + z = 24 \\ 2x + 5y + 2z = -14 \\ x - 4y + 3z = 26 \end{cases} \][/tex]

To solve this system, we need to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations simultaneously. Here's a step-by-step solution:

### Step 1: Write the equations in matrix form
We can express the system of equations in matrix form as [tex]\( AX = B \)[/tex], where:
- [tex]\( A \)[/tex] is the coefficient matrix
- [tex]\( X \)[/tex] is the column matrix of variables [tex]\( x, y, z \)[/tex]
- [tex]\( B \)[/tex] is the column matrix of constants

[tex]\[ A = \begin{pmatrix} 5 & -2 & 1 \\ 2 & 5 & 2 \\ 1 & -4 & 3 \\ \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix}, \quad B = \begin{pmatrix} 24 \\ -14 \\ 26 \\ \end{pmatrix} \][/tex]

### Step 2: Solve the matrix equation [tex]\( AX = B \)[/tex]
To find [tex]\( X \)[/tex], we need to solve the matrix equation for [tex]\( X \)[/tex]. This is done by calculating the inverse of the matrix [tex]\( A \)[/tex] and then multiplying it by the matrix [tex]\( B \)[/tex]:

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

### Step 3: Compute the result
Performing the matrix calculations, we obtain:

[tex]\[ X = \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} 2.63636364 \\ -4.54545455 \\ 1.72727273 \\ \end{pmatrix} \][/tex]

### Conclusion
Thus, the solution to the system of equations is:

[tex]\[ \boxed{x = 2.63636364, \quad y = -4.54545455, \quad z = 1.72727273} \][/tex]

These values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] satisfy all three given equations.