Question

If two equations are in the below mentioned form,
a1x+b1y+c1z=0.....(1)
a2x+b2y+c2z=0...(2)
what are values of x,y,z?
Please answer if anyone can.This answer is necessary for making a program to solve such equation for my practice.

Answer 1

Since you have two equations with three unknowns, you will need to introduce a parameter \(t\in\Bbb{R}\) into you solution. So for instance, suppose that \(z=t\) is the parameter. Then the system of equations boils down to \[\large \left\{\begin{aligned}a_1x+b_1y&=-c_1t\\ a_2x+b_2y&= -c_2t\end{aligned}\right.\]which, in matrix form, looks as follows:\[\large \begin{bmatrix}a_1 & b_1\\ a_2 & b_2\end{bmatrix} \begin{bmatrix}x\\ y\end{bmatrix} = -t\begin{bmatrix}c_1\\c_2\end{bmatrix}\](since it's easier to program things if they're in matrix form.) Thus, if it's possible to solve this system of equations (in matrix form, this means that the matrix is invertible), you'll find x and y in terms of the parameter t as well; furthemore, the solution will look something like (x(t),y(t),t). If you aren't using matrices in your program, then the best way to solve the system of equations would be via Gaussian elimination.

I hope this makes sense! :-)

Answer 2

Gaussian elimination? I don't know about it .I am a student ,so,programing is new to me.Still now our instructor have not given teach about matrix.Can YOU SAY ANOTHER way of solving?

Answer 3

What you could try to do instead is solve the system of equations by elimination or substitution (this would be a step down from the Gaussian elimination). If you do it by substitution, for example, you could solve the first equation for x and get \(\large x=\dfrac{-c_1t - b_1y}{a_1}\) and then substitute this into the second equation to get \(\large a_2\left(\dfrac{-c_1t - b_1y}{a_1}\right) + b_2 y = -c_2 t\) and then solve that resulting equation for \(y\).

Does this way look more familiar to you? :-)

Answer 4

When you do it this way, you'll need to be careful about dividing by zero, etc.

Answer 5

I will try according to your guidelines ,thank you .