Certainly! Let's solve the given system of linear equations using matrix methods. The system of equations is:
[tex]\[
\left\{\begin{array}{l}
-x - 7y - z = -19 \\
4x + 4y + 4z = 4 \\
2x + y + 6z = 7
\end{array}\right.
\][/tex]
### Step-by-Step Solution:
1. Write the system in matrix form:
The system of equations can be represented in the form [tex]\(AX = B\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(X\)[/tex] is the column vector of variables, and [tex]\(B\)[/tex] is the column vector of constants.
[tex]\[
A = \begin{pmatrix}
-1 & -7 & -1 \\
4 & 4 & 4 \\
2 & 1 & 6
\end{pmatrix}
, \quad X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
, \quad B = \begin{pmatrix}
-19 \\
4 \\
7
\end{pmatrix}
\][/tex]
2. Find the inverse of the coefficient matrix [tex]\(A\)[/tex] (if it exists):
For our purposes, it's sufficient to know that the matrix [tex]\(A\)[/tex] is invertible, and we will use the inverse to find [tex]\(X\)[/tex].
3. Multiply both sides of the equation [tex]\(AX = B\)[/tex] by the inverse of [tex]\(A\)[/tex]:
[tex]\[
A^{-1}AX = A^{-1}B
\][/tex]
Since [tex]\(A^{-1}A\)[/tex] is the identity matrix [tex]\(I\)[/tex], this simplifies to:
[tex]\[
IX = A^{-1}B \quad \text{or simply} \quad X = A^{-1}B
\][/tex]
4. Calculate [tex]\(X\)[/tex]:
Using the result obtained from computation:
[tex]\[
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
= \begin{pmatrix}
-4 \\
3 \\
2
\end{pmatrix}
\][/tex]
### Verification:
To ensure the solution is correct, it's good practice to substitute [tex]\(x = -4\)[/tex], [tex]\(y = 3\)[/tex], and [tex]\(z = 2\)[/tex] back into the original equations:
1. For [tex]\( -x - 7y - z = -19 \)[/tex]:
[tex]\[
-(-4) - 7(3) - 1(2) = 4 - 21 - 2 = -19
\][/tex]
2. For [tex]\( 4x + 4y + 4z = 4 \)[/tex]:
[tex]\[
4(-4) + 4(3) + 4(2) = -16 + 12 + 8 = 4
\][/tex]
3. For [tex]\( 2x + y + 6z = 7 \)[/tex]:
[tex]\[
2(-4) + 3 + 6(2) = -8 + 3 + 12 = 7
\][/tex]
The solution satisfies all the original equations, confirming that our solution is correct.
Thus, the solution to the system is:
[tex]\[
x = -4, \quad y = 3, \quad z = 2
\][/tex]
