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.
