Answer :
Let's analyze each step that Galena took when solving the system of equations:
Given system:
[tex]\[ \left\{\begin{aligned} 3x + 2y + 3z &= 5 \quad \text{(Equation 1)} \\ 7x + y + 7z &= -1 \quad \text{(Equation 2)} \\ 4x - 4y - z &= -3 \quad \text{(Equation 3)} \end{aligned}\right. \][/tex]
### Step 1: Multiplying Equation (3) by 3 and adding to Equation (1)
First, we multiply Equation (3) by [tex]\(3\)[/tex]:
[tex]\[ 3(4x - 4y - z) = 12x - 12y - 3z = -9 \][/tex]
Next, we add this result to Equation (1):
[tex]\[ (3x + 2y + 3z) + (12x - 12y - 3z) = 5 + (-9) \][/tex]
Combining like terms, we get:
[tex]\[ (3x + 12x) + (2y - 12y) + (3z - 3z) = 5 - 9 \][/tex]
[tex]\[ 15x - 10y + 0z = -4 \][/tex]
So, after Step 1, the new form of the equation derived from Equation (1) and Equation (3) is:
[tex]\[ 15x - 10y = -4 \][/tex]
(Let's call this Equation [tex]\(1'\)[/tex])
### Step 2: Multiplying Equation (3) by -7 and adding to Equation (2)
Next, we multiply Equation (3) by [tex]\(-7\)[/tex]:
[tex]\[ -7(4x - 4y - z) = -28x + 28y + 7z = 21 \][/tex]
Now, we add this result to Equation (2):
[tex]\[ (7x + y + 7z) + (-28x + 28y + 7z) = -1 + 21 \][/tex]
Combining like terms, we get:
[tex]\[ (7x - 28x) + (y + 28y) + (7z + 7z) = -1 + 21 \][/tex]
[tex]\[ -21x + 29y + 14z = 20 \][/tex]
So, after Step 2, the new form of the equation derived from Equation (2) and Equation (3) is:
[tex]\[ -21x + 29y + 14z = 20 \][/tex]
(Let's call this Equation [tex]\(2'\)[/tex])
### Analysis of Statements Explaining Galena's Mistake
We need to check if Galena made any errors in her steps:
1. She added Equation (1) instead of Equation (2) in Step 1:
- This would mean adding Equation (1) to the modified Equation (3).
[tex]\[ (3x + 2y + 3z) + (-28x + 28y + 7z) = 5 + 21 \][/tex]
[tex]\[ (-25x + 30y + 10z = 26) \][/tex]
This result does not match the steps Galena followed.
2. She did not multiply Equation (3) in Step 1 by the correct value:
- The correct multiplication was [tex]\(3\)[/tex].
[tex]\[ 3 \cdot (4x - 4y - z) = 12x - 12y - 3z \][/tex]
[tex]\[ (3x + 2y + 3z) + (12x - 12y - 3z) = -4 \][/tex]
The multiplication and addition were correctly done.
3. She did not multiply Equation (3) in Step 2 by the correct value:
- The correct multiplication was [tex]\(-7\)[/tex].
[tex]\[ -7 \cdot (4x - 4y - z) = -28x + 28y + 7z \][/tex]
[tex]\[ (7x + y + 7z) + (-28x + 28y + 7z) = 20 \][/tex]
The multiplication and addition were correctly done.
4. She added Equation (2) instead of Equation (1) in Step 2:
- This would mean adding Equation (2) to the modified Equation (3).
[tex]\[ (7x + y + 7z) + (12x - 12y - 3z) = -1 - 9 \][/tex]
[tex]\[ 19x - 11y + 4z = -10 \][/tex]
This result matches the incorrect addition, indicating a mistake.
### Conclusion
Galena's steps were correct. Therefore:
The correct statement is:
- None of the given statements are true as there was no mistake in Galena's method. Both steps were correctly executed including the multiplications and additions with the respective equations.
Given system:
[tex]\[ \left\{\begin{aligned} 3x + 2y + 3z &= 5 \quad \text{(Equation 1)} \\ 7x + y + 7z &= -1 \quad \text{(Equation 2)} \\ 4x - 4y - z &= -3 \quad \text{(Equation 3)} \end{aligned}\right. \][/tex]
### Step 1: Multiplying Equation (3) by 3 and adding to Equation (1)
First, we multiply Equation (3) by [tex]\(3\)[/tex]:
[tex]\[ 3(4x - 4y - z) = 12x - 12y - 3z = -9 \][/tex]
Next, we add this result to Equation (1):
[tex]\[ (3x + 2y + 3z) + (12x - 12y - 3z) = 5 + (-9) \][/tex]
Combining like terms, we get:
[tex]\[ (3x + 12x) + (2y - 12y) + (3z - 3z) = 5 - 9 \][/tex]
[tex]\[ 15x - 10y + 0z = -4 \][/tex]
So, after Step 1, the new form of the equation derived from Equation (1) and Equation (3) is:
[tex]\[ 15x - 10y = -4 \][/tex]
(Let's call this Equation [tex]\(1'\)[/tex])
### Step 2: Multiplying Equation (3) by -7 and adding to Equation (2)
Next, we multiply Equation (3) by [tex]\(-7\)[/tex]:
[tex]\[ -7(4x - 4y - z) = -28x + 28y + 7z = 21 \][/tex]
Now, we add this result to Equation (2):
[tex]\[ (7x + y + 7z) + (-28x + 28y + 7z) = -1 + 21 \][/tex]
Combining like terms, we get:
[tex]\[ (7x - 28x) + (y + 28y) + (7z + 7z) = -1 + 21 \][/tex]
[tex]\[ -21x + 29y + 14z = 20 \][/tex]
So, after Step 2, the new form of the equation derived from Equation (2) and Equation (3) is:
[tex]\[ -21x + 29y + 14z = 20 \][/tex]
(Let's call this Equation [tex]\(2'\)[/tex])
### Analysis of Statements Explaining Galena's Mistake
We need to check if Galena made any errors in her steps:
1. She added Equation (1) instead of Equation (2) in Step 1:
- This would mean adding Equation (1) to the modified Equation (3).
[tex]\[ (3x + 2y + 3z) + (-28x + 28y + 7z) = 5 + 21 \][/tex]
[tex]\[ (-25x + 30y + 10z = 26) \][/tex]
This result does not match the steps Galena followed.
2. She did not multiply Equation (3) in Step 1 by the correct value:
- The correct multiplication was [tex]\(3\)[/tex].
[tex]\[ 3 \cdot (4x - 4y - z) = 12x - 12y - 3z \][/tex]
[tex]\[ (3x + 2y + 3z) + (12x - 12y - 3z) = -4 \][/tex]
The multiplication and addition were correctly done.
3. She did not multiply Equation (3) in Step 2 by the correct value:
- The correct multiplication was [tex]\(-7\)[/tex].
[tex]\[ -7 \cdot (4x - 4y - z) = -28x + 28y + 7z \][/tex]
[tex]\[ (7x + y + 7z) + (-28x + 28y + 7z) = 20 \][/tex]
The multiplication and addition were correctly done.
4. She added Equation (2) instead of Equation (1) in Step 2:
- This would mean adding Equation (2) to the modified Equation (3).
[tex]\[ (7x + y + 7z) + (12x - 12y - 3z) = -1 - 9 \][/tex]
[tex]\[ 19x - 11y + 4z = -10 \][/tex]
This result matches the incorrect addition, indicating a mistake.
### Conclusion
Galena's steps were correct. Therefore:
The correct statement is:
- None of the given statements are true as there was no mistake in Galena's method. Both steps were correctly executed including the multiplications and additions with the respective equations.
