Question

Algebra Help

Consider the system shown:
x - 3y + z = 6
x + 4y - 2z = 9

How many solutions does this system have?

Make a conjecture about the minimun number of equations that a linear system in n variables can have when there is exactly one solution.

Answer 1

since \(\Large{\frac{a_1}{a_2} \ne \frac{b_1}{b_2} \ne \frac{c_1}{c_2}}\)
therefore the system possesses a unique solution

Answer 2

Also to find the solution of a linear system consisting of n variables u need minimum of n equations

Answer 3

I don't get the first part D:

Answer 4

In order to find a solution, you need 1 equation for each unknown.
Since there are 3 unknowns you need a third equation.

Answer 5

Underdetermined system...more variables than equations. Can have either infinite number of solutions or zero solutions.

Answer 6

Yes, I get the part about n equations for n variables

Answer 7

\[a_1x_1+b_1y_1+c_1z_1=d_1\]\[a_2x_2+b_2y_2+c_2z_2=d_2\]

\[\frac{a_1}{a_2} \ne \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \] means that the system has a unique solution

Answer 8

\[\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\]means that the system possesses infinite number of solutions

Answer 9

\[\frac{a_1}{a_2}=\frac{b_1}{b_2} \ne \frac{c_1}{c_2}\]means that the system possesses no solutions

Answer 10

Thanks O_O that's weird

Answer 11

Except if you have 1 equation more. It can be have just one solution.

Answer 12

What does that mean? @Kevin

Answer 13

By the way, Kevin
Even it has one more equation, it is not a condition to has a unique solution, may be none

Answer 14

This is linear equation with 3 variables, you need 3 equations to find each variable values where there is exactly one solution.