To solve the problem, we need to find the coordinates of points L and M from the equations \( AX = C \) and \( BX = D \), and then find their reflections \( L' \) and \( M' \) in the plane \( x + y + z = 9 \).
### Step 1: Finding the coordinates of point L from \( AX = C \)
Given:
- \( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 1 & -1 & 1 \end{pmatrix} \)
- \( C = \begin{pmatrix} 14 \\ 12 \\ 2 \end{pmatrix} \)
We need to solve the equation \( AX = C \).
This expands to the following system of equations:
1. \( x + 2y + 3z = 14 \) (Equation 1)
2. \( 4x + y + 2z = 12 \) (Equation 2)
3. \( x - y + z = 2 \) (Equation 3)
#### Solving the equations:
From Equation 3:
\[ z = 2 - x + y \]
Substituting \( z \) into Equations 1 and 2:
**Substituting into Equation 1:**
\[ x + 2y + 3(2 - x + y) = 14 \]
\[ x + 2y + 6 - 3x + 3y = 14 \]
\[ -2x + 5y + 6 = 14 \]
\[ -2x + 5y = 8 \]
\[ 2x - 5y = -8 \] (Equation 4)
**Substituting into Equation 2:**
\[ 4x + y + 2(2 - x + y) = 12 \]
\[ 4x + y + 4 - 2x + 2y = 12 \]
\[ 2x + 3y + 4 = 12 \]
\[ 2x + 3y = 8 \] (Equation 5)
Now we have two equations (4 and 5):
1. \( 2x - 5y = -8 \) (Equation 4)
2. \( 2x + 3y = 8 \) (Equation 5)
Subtract Equation 4 from Equation 5:
\[ (2x + 3y) - (2x - 5y) = 8 + 8 \]
\[ 8y = 16 \]
\[ y = 2 \]
Substituting \( y = 2 \) back into Equation 5:
\[ 2x + 3(2) = 8 \]
\[ 2x + 6 = 8 \]
\[ 2x = 2 \]
\[ x = 1 \]
Now substituting \( x = 1 \) and \( y = 2 \) into Equation 3 to find \( z \):
\[ z = 2 - 1 + 2 = 3 \]
Thus, the coordinates of point L are:
\[ L(1, 2, 3) \]
### Step 2: Finding the coordinates of point M from \( BX = D \)
Given:
- \( B = \begin{pmatrix} 2 & 1 & 3 \\ 4 & 1 & -1 \\ 2 & 2 & 3 \end{pmatrix} \)
- \( D = \begin{pmatrix} 13 \\ 11 \\ 14 \end{pmatrix} \)
We need to solve the equation \( BX = D \).
This expands to the following system of equations:
1. \( 2x + y + 3z = 13 \) (Equation 6)
2. \( 4x + y - z = 11 \) (Equation 7)
3. \( 2x + 2y + 3z = 14 \) (Equation 8)
#### Solving the equations:
From Equation 7:
\[ z = 4x + y - 11 \]
Substituting \( z \) into Equations 6 and 8:
**Substituting into Equation 6:**
\[ 2x + y + 3(4x + y - 11) = 13 \]
\[ 2x + y + 12x + 3y - 33 = 13 \]
\[ 14x + 4y - 33 = 13 \]
\[ 14x + 4y = 46 \]
\[ 7x + 2y = 23 \] (Equation 9)
**Substituting into Equation 8:**
\[ 2x + 2y + 3(4x + y - 11) = 14 \]
\[ 2x + 2y + 12x + 3y - 33 = 14 \]
\[ 14x + 5y - 33 = 14 \]
\[ 14x + 5y = 47 \] (Equation 10)
Now we have two equations (9 and 10):
1. \( 7x + 2y = 23 \) (Equation 9)
2. \( 14x + 5y = 47 \) (Equation 10)
From Equation 9, express \( y \):
\[ 2y = 23 - 7x \]
\[ y = \frac{23 - 7x}{2} \]
Substituting into Equation 10:
\[ 14x + 5\left(\frac{23 - 7x}{2}\right) = 47 \]
\[ 14x + \frac{115 - 35x}{2} = 47 \]
Multiply through by 2 to eliminate the fraction:
\[ 28x + 115 - 35x = 94 \]
\[ -7x + 115 = 94 \]
\[ -7x = -21 \]
\[ x = 3 \]
Substituting \( x = 3 \) back into Equation 9:
\[ 7(3) + 2y = 23 \]
\[ 21 + 2y = 23 \]
\[ 2y = 2 \]
\[ y = 1 \]
Now substituting \( x = 3 \) and \( y = 1 \) into Equation 7 to find \( z \):
\[ z = 4(3) + 1 - 11 = 12 - 11 = 1 \]
Thus, the coordinates of point M are:
\[ M(3, 1, 1) \]
### Step 3: Finding reflections \( L' \) and \( M' \)
The plane equation is \( x + y + z = 9 \).
**Finding reflection \( L' \) of point L:**
Coordinates of L: \( (1, 2, 3) \)
1. Calculate the point on the plane:
\[ x + y + z = 1 + 2 + 3 = 6 \]
The distance to the plane is \( 9 - 6 = 3 \).
2. The reflection point \( L' \) will be:
\[ L' = (1, 2, 3) + 2(3, 3, 3) = (1 + 3, 2 + 3, 3 + 3) = (4, 5, 6) \]
**Finding reflection \( M' \) of point M:**
Coordinates of M: \( (3, 1, 1) \)
1. Calculate the point on the plane:
\[ x + y + z = 3 + 1 + 1 = 5 \]
The distance to the plane is \( 9 - 5 = 4 \).
2. The reflection point \( M' \) will be:
\[ M' = (3, 1, 1) + 2(4, 4, 4) = (3 + 4, 1 + 4, 1 + 4) = (7, 5, 5) \]
### Final Coordinates:
- \( L(1, 2, 3) \)
- \( M(3, 1, 1) \)
- \( L'(4, 5, 6) \)
- \( M'(7, 5, 5) \)
