Determine the intersection of the three planes with the following system of equations:
x - y + 2z = 0
3x + 2y - z= -5
4x + 2y - 2z = -6
multiply the second equation by 2 to get rid of z
x - y + 2z = 0
6x + 4y -2z= -10
4x + 2y - 2z = -6
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7 x + 3 y = -10
2 x + 2 y = - 4
multiply first by 2 and second by 3
14 x + 6 y = - 20
6 x + 6 y = -12
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8 x = - 8
x = -1
now go back up and get y and z
thanks
To find the intersection points of the three planes, you need to solve the system of equations simultaneously. Here's how you can do it:
Step 1: Write down the system of equations:
x - y + 2z = 0 ---(1)
3x + 2y - z = -5 ---(2)
4x + 2y - 2z = -6 ---(3)
Step 2: Use any method to solve the system of equations (e.g., substitution, elimination).
Let's use the method of elimination:
Multiply equation (1) by 2:
2x - 2y + 4z = 0 ---(4)
Subtract equation (2) from equation (4):
2x - 2y + 4z - (3x + 2y - z) = 0 - (-5)
Simplifying the equation gives:
2x - 2y + 4z - 3x - 2y + z = 5
-x - 4y + 5z = 5 ---(5)
Now, subtract equation (3) from equation (4):
2x - 2y + 4z - (4x + 2y - 2z) = 0 - (-6)
Simplifying the equation gives:
2x - 2y + 4z - 4x - 2y + 2z = 6
-2x - 4y + 6z = 6 ---(6)
Step 3: Now you have a system of two equations:
-x - 4y + 5z = 5 ---(5)
-2x - 4y + 6z = 6 ---(6)
Step 4: Apply the method of elimination to solve equations (5) and (6).
Multiply equation (5) by 2:
-2x - 8y + 10z = 10 ---(7)
Subtract equation (6) from equation (7):
-2x - 8y + 10z - (-2x - 4y + 6z) = 10 - 6
Simplifying the equation gives:
-8y + 4z = 4 ---(8)
Step 5: Now you have the system of equations:
-8y + 4z = 4 ---(8)
-2x - 4y + 6z = 6 ---(6)
You can solve this system by either substitution or elimination. However, in this case, the equation (8) will provide a solution directly.
Step 6: Solve equation (8) for y:
-8y + 4z = 4
Rearrange the equation:
-8y = -4z + 4
y = (4z - 4)/8
y = (z - 1)/2 ---(9)
Step 7: Substitute equation (9) into equation (6) to solve for x and z:
-2x - 4((z - 1)/2) + 6z = 6
Simplifying the equation gives:
-2x - 2(z - 1) + 6z = 6
-2x - 2z + 2 + 6z = 6
-2x + 4z = 4
Rearrange the equation:
-2x = -4z + 4
2x = 4z - 4
x = (4z - 4)/2
x = 2z - 2 ---(10)
Step 8: Substitute equations (9) and (10) back into equation (1) to solve for z:
(2z - 2) - (z - 1) + 2z = 0
Simplifying the equation gives:
2z - 2 - z + 1 + 2z = 0
3z - 1 = 0
3z = 1
z = 1/3
Step 9: Now that you have the value of z, substitute it back into equations (9) and (10) to find the values of y and x, respectively:
From equation (9):
y = (z - 1)/2
y = (1/3 - 1)/2
y = (-2/3)
From equation (10):
x = 2z - 2
x = 2(1/3) - 2
x = (-4/3)
Therefore, the solution to the system of equations is x = -4/3, y = -2/3, and z = 1/3.
Hence, the intersection point of the three planes is (-4/3, -2/3, 1/3).