To find the solution to the system of linear equations given by:
[tex]\[
\left\{\begin{array}{l}
4x + 3y - z = -6 \\
6x - y + 3z = 12 \\
8x + 2y + 4z = 6
\end{array}\right.
\][/tex]
let's verify the solutions step-by-step for consistency. Here are the steps to determine the correct values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
### Step 1: Verify the Equation 1
Substituting the given values into the first equation:
[tex]\[ 4x + 3y - z = -6 \][/tex]
Let's test the solutions:
Option 1: [tex]\( x = 1, y = -3, z = -1 \)[/tex]
[tex]\[
4(1) + 3(-3) - (-1) = 4 - 9 + 1 = -4 \quad (\text{Not equal to } -6)
\][/tex]
Option 2: [tex]\( x = 1, y = -3, z = 1 \)[/tex]
[tex]\[
4(1) + 3(-3) - 1 = 4 - 9 - 1 = -6 \quad (\text{Correct})
\][/tex]
Option 3: [tex]\( x = 1, y = 3, z = 19 \)[/tex]
[tex]\[
4(1) + 3(3) - 19 = 4 + 9 - 19 = -6 \quad (\text{Correct})
\][/tex]
Option 4: [tex]\( x = 1, y = 3, z = -2 \)[/tex]
[tex]\[
4(1) + 3(3) + 2 = 4 + 9 + 2 = 15 \quad (\text{Not equal to } -6)
\][/tex]
### Step 2: Verify the Equation 2
Substituting the given values into the second equation:
[tex]\[ 6x - y + 3z = 12 \][/tex]
Option 1: [tex]\( x = 1, y = -3, z = -1 \)[/tex]
[tex]\[
6(1) - (-3) + 3(-1) = 6 + 3 - 3 = 6 \quad (\text{Not equal to } 12)
\][/tex]
Option 2: [tex]\( x = 1, y = -3, z = 1 \)[/tex]
[tex]\[
6(1) - (-3) + 3(1) = 6 + 3 + 3 = 12 \quad (\text{Correct})
\][/tex]
Option 3: [tex]\( x = 1, y = 3, z = 19 \)[/tex]
[tex]\[
6(1) - (3) + 3(19) = 6 - 3 + 57 = 60 \quad (\text{Not equal to } 12)
\][/tex]
Option 4: [tex]\( x = 1, y = 3, z = -2 \)[/tex]
[tex]\[
6(1) - 3 + 3(-2) = 6 - 3 - 6 = -3 \quad (\text{Not equal to } 12)
\][/tex]
### Step 3: Verify the Equation 3
Substituting the given values into the third equation:
[tex]\[ 8x + 2y + 4z = 6 \][/tex]
Option 1: [tex]\( x = 1, y = -3, z = -1 \)[/tex]
[tex]\[
8(1) + 2(-3) + 4(-1) = 8 - 6 - 4 = -2 \quad (\text{Not equal to } 6)
\][/tex]
Option 2: [tex]\( x = 1, y = -3, z = 1 \)[/tex]
[tex]\[
8(1) + 2(-3) + 4(1) = 8 - 6 + 4 = 6 \quad (\text{Correct})
\][/tex]
Option 3: [tex]\( x = 1, y = 3, z = 19 \)[/tex]
[tex]\[
8(1) + 2(3) + 4(19) = 8 + 6 + 76 = 90 \quad (\text{Not equal to } 6)
\][/tex]
Option 4: [tex]\( x = 1, y = 3, z = -2 \)[/tex]
[tex]\[
8(1) + 2(3) + 4(-2) = 8 + 6 - 8 = 6 \quad (\text{Correct})
\][/tex]
### Conclusion
The solution that satisfies all three equations is:
[tex]\[
(x, y, z) = (1, -3, 1)
\][/tex]
Thus, the correct solution to the system of linear equations is:
[tex]\[
x = 1, \quad y = -3, \quad z = 1
\][/tex]
