3: Point Intersection
In the simplest case, three planes intersect at a unique point
Unless two planes are parallel, they intersect along a line. For example, the line between the green plane and the red plane on the right.
When a third plane is added into the mix, it often intersects the first two at a single point. For example, the purple plane intersects the line at the point in the 'corner' near the middle.
An effective method for finding the intersection point is elimination.
Given three plane equations, write two new equations each of which has the same variable eliminated.
From these two new equations (now only involving the same two variables) using elimination again to find their values.
Finally, substitute back to find the values.
Example
Find the intersection of the planes
(eq1) x + y + 2z = 6
(eq2) 2x + 3y + 5z = 10
(eq3) 3x + 7y + 7z = 1
Solution
Use elimination on equations (eq1) and (eq2): (eq2) - 2(eq1)
(eq4) 0x + 1y + 1z = -2
Use elimination on equations (eq1) and (eq3): (eq3) - 3(eq1)
(eq5) 0x + 4y + 1z = -17
Now use elimination on equations (eq4) and (eq5): (eq5) - 4(eq4)
(eq6) 0x + 0y - 3z = -9
From this, we can see that z = 3.
Substituting back into (eq4) [or into (eq5)] we find y = -5.
Substituting back into (eq1) [or (eq2) or (eq3)] we find x = 5.
The only point on all three planes is (x,y,z) = (5,-5,3).
We can check that these points are on all three planes by substituting back into the equations for the three planes.
When making a list of equations, it can be useful to name or number them so they can be referred to later.
In most of this site, we use (eq1), (eq2) etc. to name each equation.
An alternative approach is to number the equations and put the number in a circle.
It's important not to get the numbers from the equations mixed up with the numbers in the equations; the brackets or circles help.
With a calculator
Although doing this with algebra is important, the good news is that your calculator can find a unique solution (and so can many computer software packages, websites and even Excel).
The only requirement is that the equations are in general form, Ax + By + Cz = D.
On a Casio fx-9860 SD (or similar), go to (MENU) and choose EQUA.
Select "Simultaneous": (F1)
Select "Number of Unknowns" (F2) for 3 unknowns
Enter the coefficients for the equations (12 numbers in rows).
Select Solve (F1)
The solution will be displayed, as long as it is unique.
Otherwise, "Ma Error" is displayed.
Real World Example
Adam wants to buy some coffee from a local supplier. The company has coffee from Colombia, Ecuador and Kenya, which they sell by the kilogram.
Colombian coffee costs $12 per kilogram, Ecuadorian coffee costs $18 per kilogram and Kenyan coffee costs $25 per kilogram.
Adam wants to buy 20kg of coffee.
Adam wants to spend $400 on coffee.
Adam wants to buy twice as much Kenyan coffee as Colombian coffee.
Solution
Let x, y and z be the weights (in kilograms) of coffee purchased from Colombia, Ecuador and Kenya respectively.
For the total weight
1x + 1y + 1z = 20
For the total cost
12x + 18y + 25z = 400
For the ratio of Kenyan and Colombian coffee
2x = z
Notice that this equation can be written in general form as
2x + 0y - 1z = 0
Putting these equations into a graphics calculator, we get the output
(x,y,z) = (5,5,10)
Putting this in context, Adam should by 5kg of Colombian coffe, 5kg of Ecuadorian coffee and 10kg of Kenya coffee to satisfy his conditions.
Casio calculator input for solving the problem.
One of the most common mistakes made in problems like this one is to write the wrong third equation for the ratio: that is, to use x = 2z.
If you're not sure if you've got it right, try checking to see if a specific situation fits the equation. In this example, twice as much Kenyan as Colombian coffee could be z=4 and x=2. This fits with the equation 2x = z.
Exercises
For each systems of equations, find the (unique) solution.
Try to solve with algebra first, and then check with your calculator.
A:
x + y + 3z = 11
2x + y + z = 13
3x + y + 4z = 18
B:
3x + y + z = 3
3x + 2y + 3z = 6
5x + 7y + 4z = 11
C:
10x + y = 61
10y + z = 18
10z + x = 86
Investigation
In Adam's coffee problem above, suppose Adam had the same budget, and same third requirement about the ratio of Kenyan and Colombian coffee.
What is the least total weight of coffee he could buy, and the most?