Answer:
Explanation:
a) The system of planes is represented by the following equations:
3x + 2y - z = -2
2x + y - 2z = 7
2x - 3y + 4z = -3
To solve this system, we can use various methods such as substitution, elimination, or matrix methods. Here, we'll use the elimination method to find the solution:
Step 1: Multiply the second equation by 2 and the third equation by 3 to create a cancellation:
3x + 2y - z = -2
4x + 2y - 4z = 14
6x - 9y + 12z = -9
Step 2: Subtract the first equation from the second equation and the first equation from the third equation:
4x + 2y - 4z - (3x + 2y - z) = 14 - (-2)
6x - 9y + 12z - (3x + 2y - z) = -9 - (-2)
Simplifying, we get:
x - 3z = 16 (Equation 4)
3x - 11y + 13z = -7 (Equation 5)
Step 3: Multiply Equation 4 by 3 and add it to Equation 5:
3(x - 3z) + (3x - 11y + 13z) = 16(3) - 7
Simplifying, we have:
6x - 2y + 4z = 37 (Equation 6)
Step 4: Now we have the following two equations:
6x - 2y + 4z = 37 (Equation 6)
2x + y - 2z = 7 (Equation 2)
Step 5: We can now solve this system of equations. One way is to multiply Equation 2 by 2 and add it to Equation 6:
2(2x + y - 2z) + (6x - 2y + 4z) = 2(7) + 37
Simplifying, we get:
10x = 51
Dividing both sides by 10, we find:
x = 5.1
Step 6: Substituting the value of x into Equation 2, we can solve for y:
2(5.1) + y - 2z = 7
Simplifying, we have:
y - 2z = -3.2 (Equation 7)
Step 7: Substituting the values of x and y into Equation 6, we can solve for z:
6(5.1) - 2y + 4z = 37
Simplifying, we get:
4z = 1.8
Dividing both sides by 4, we find:
z = 0.45
Therefore, the solution to the system of planes is x = 5.1, y = -3.2, and z = 0.45.
b) The system of planes is represented by the following equations:
3x - 4y + 2z = 1
6x - 8y + 4z = 10
15x - 20y + 10z = -3
To solve this system, we'll again use the elimination method:
Step 1: Multiply the first equation by 2 and the second equation by 5 to create a cancellation:
6x - 8y + 4z = 2
30x -
I hope this is right!! :)