Post on 18-May-2015
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3 dimensional space can be represented by a Cartesian grid with 3 axes:
x, y and z
In two dimensional space a line is represented by an equation of the form
Ax + By + C = 0
In 3 dimensional space a plane is represented by an equation of the form
Ax + By + Cz + D = 0
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Solving a System of PlanesThe solution to a system of 3 planes has 4 possibilities:
1. They intersect at a common point and have one solution.(x, y, z)
2. They might intersect in a common line.
3. They might have no points in common (no solution).
4. The 3 equations might all represent the same plane.(the solution is all points on the plane)
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( 9, 4, 0 )Now how do you solve if your system is a system of 3 planes which have 3 variables each?
Just like with two variables you can solve by either
elimination or substitution
Only now there will be more operations involved
Use the same thought process:
1. Get to the point where you have only two variables by elimination or substitution.
2. When you are down to two variables and two equations it is just like what we were doing in the previous lesson.
3. Solve for one of the two variables.
4. Substitute in order to find the second variable in the equation that only involves two variables.
5. Now that you have the value of the two coordinates, solve for the third variable.
6. Check by substituting your values for (x, y, z)
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4x 10y 5z = 76
4x 6y 5z = 60
7y 2z = 28
1. Get to the point where you have only two variables by elimination or substitution.
2. When you are down to two variables it is just like what we were doing in the previous lesson.
3. Solve for one of the two variables.
4. Substitute in order to find the second variable in the equation that only involves two variables.
5. Now that you have the value of the two coordinates, solve for the third variable.
6. Check by substituting your values for (x, y, z)
Example
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11x 6y 7z = 74
4x + 5y + 5z = 56
5x + 5y 6z = 54
Solve this system of equations
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Exercise 25
question 1