To solve the given system of equations represented by the augmented matrix in row echelon form, follow these step-by-step instructions:
### Step 1: Write the system of equations
The augmented matrix
[tex]\[
\left|\begin{array}{cccc}
1 & -3 & 4 & 7 \\
0 & 1 & 2 & 2 \\
0 & 0 & 1 & 5
\end{array}\right|
\][/tex]
corresponds to the system of linear equations:
1. [tex]\( x - 3y + 4z = 7 \)[/tex]
2. [tex]\( y + 2z = 2 \)[/tex]
3. [tex]\( z = 5 \)[/tex]
### Step 2: Solve for [tex]\( z \)[/tex]
The third row of the matrix (or the third equation) directly gives us the value of [tex]\( z \)[/tex]:
[tex]\[ z = 5 \][/tex]
### Step 3: Substitute [tex]\( z \)[/tex] into the second equation to solve for [tex]\( y \)[/tex]
Using the value of [tex]\( z \)[/tex] from Step 2, substitute [tex]\( z = 5 \)[/tex] into the second equation:
[tex]\[ y + 2z = 2 \][/tex]
Substitute [tex]\( z = 5 \)[/tex]:
[tex]\[ y + 2(5) = 2 \][/tex]
[tex]\[ y + 10 = 2 \][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2 - 10 \][/tex]
[tex]\[ y = -8 \][/tex]
### Step 4: Substitute [tex]\( y \)[/tex] and [tex]\( z \)[/tex] into the first equation to solve for [tex]\( x \)[/tex]
Using the values of [tex]\( y \)[/tex] and [tex]\( z \)[/tex] from the previous steps, substitute [tex]\( y = -8 \)[/tex] and [tex]\( z = 5 \)[/tex] into the first equation:
[tex]\[ x - 3y + 4z = 7 \][/tex]
Substitute [tex]\( y = -8 \)[/tex] and [tex]\( z = 5 \)[/tex]:
[tex]\[ x - 3(-8) + 4(5) = 7 \][/tex]
[tex]\[ x + 24 + 20 = 7 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x + 44 = 7 \][/tex]
[tex]\[ x = 7 - 44 \][/tex]
[tex]\[ x = 11 \][/tex]
### Final Solution
The solution to the system of equations is:
[tex]\[ x = 11, \; y = -8, \; z = 5 \][/tex]
So, the solution in ordered triplet form is:
[tex]\[ (x, y, z) = (11, -8, 5) \][/tex]
