Let's examine Mark's steps and determine where he went wrong in his approach.
### Given System of Equations:
1. [tex]\( x + y + z = 2 \)[/tex]
2. [tex]\( 3x + 2y + z = 8 \)[/tex]
3. [tex]\( 4x - y - 7z = 16 \)[/tex]
### Step 1: He multiplies equation (1) by 7 and adds it to equation (3).
- Multiply equation (1) by 7:
[tex]\[ 7(x + y + z) = 7 \cdot 2 \][/tex]
[tex]\[ 7x + 7y + 7z = 14 \][/tex]
- Add the result to equation (3):
[tex]\[ (7x + 7y + 7z) + (4x - y - 7z) = 14 + 16 \][/tex]
[tex]\[ 7x + 7y + 7z + 4x - y - 7z = 30 \][/tex]
[tex]\[ 11x + 6y = 30 \][/tex]
### Step 2: He multiplies equation (3) by 2 and adds it to equation (2).
- Multiply equation (3) by 2:
[tex]\[ 2(4x - y - 7z) = 2 \cdot 16 \][/tex]
[tex]\[ 8x - 2y - 14z = 32 \][/tex]
- Add the result to equation (2):
[tex]\[ (8x - 2y - 14z) + (3x + 2y + z) = 32 + 8 \][/tex]
[tex]\[ 8x - 2y - 14z + 3x + 2y + z = 40 \][/tex]
[tex]\[ 11x - 13z = 40 \][/tex]
### Identify Mark's Mistake:
Let's analyze the result and method:
- Elimination Goal: In solving systems of equations, the goal is usually to eliminate one variable step by step. Mark did not focus on consistently eliminating the same variable in both steps.
- Step 1 Observation: By performing the operations in Step 1, Mark got [tex]\( 11x + 6y = 30 \)[/tex].
- Step 2 Observation: In Step 2, he got [tex]\( 11x - 13z = 40 \)[/tex].
Since Mark's steps led to inconsistent elimination, meaning he did not focus on removing the same variable in each step, he's not properly narrowing down the solution systematically. The right approach should have been to ensure the elimination of the same variable in both steps to proceed towards a triangular form or row-echelon form.
### Conclusion
The statement that explains Mark's mistake is:
He did not eliminate the same variables in steps 1 and 2.
