Solving 3 Variables with 2 Points: Can We Do It?

[mubashirmansoor]

can we do this?

Can we solve 3 variables with 2 points? like this one

4a+2b+c=4
a+b+c=1

can we get the numerical values of all 3 variables?

 

[HallsofIvy]

No. If you think of a, b, c as the x,y,z components of a point in 3 dimensional space, the 2 equations are equations of planes. The two equations will be satisfied for points where the two planes intersect. Two planes either are parallel and so don't intersect (so there is no solution) or intersect in a line (so there are an infinite number of solutions). But there cannot be a case where two planes intersect in a single point.

 

[murshid_islam]

4a+2b+c=4
a+b+c=1

subtract the 2nd eqn. from the 1st one and you get:
3a + b = 3
or, b = 3 - 3a

here putting different values of a, you can get different values of b. and then putting those values of a and b in either one of the original equations, you can get the value of c. thus, you can get and infinite number of solutions.

for example, if a = 1, then
b = 3-3a = 3-3*1 = 0 and
c = 1-a-b = 1-1-0 = 0
so, one solution is (a,b,c) = (1,0,0)

 

[matt grime]

By inspection you can see there is no unique answer.

 

[murshid_islam]

By inspection you can see there is no unique answer.

of course we can. i was not sure whether mubashirmansoor wanted only unique answers. i thought he might have wanted a number of solutions (like the solutions of a diophantine equations).