Question

Use gaussian elimination for the following problem

Answer 1

8x-5y = 20

-16x+10y = -40

Answer 2

I multiply the top equation by two and then this is what happens.

-2(8x-5y = 20) = -16x+10y = 40
-16x+10y = 40

i get 0 = 0.

How can I find a solution to this can anyone help?

Answer 3

please help

Answer 4

@mathstudent55

Answer 5

the equations are linearly dependent. you can assign a parameter for either variable. e.g let \[y=t,\quad t\in \mathbb{R}\]

Answer 6

there are infinity solutions, btw to use Gaussian elimination you should end up with a matrix in row reduced echelon form.

Answer 7

*infinitely many

Answer 8

\[\left[\begin{matrix}8 & -5 \\ -16 & 10\end{matrix}\right] = 80-80 = 0\]

Answer 9

so essentially I can substitute a variable say y = t and then chose it to be whatever I want?

Answer 10

But I guess this is resting on the fact that these two equations are the same.

Answer 11

I just realized that one is a multiple of the other.

Answer 12

@Redcan how would I proceed?

Answer 13

@Clalgee

Answer 14

So because one is a multiple of another, there can be infinitely many solutions.

Answer 15

yep, infinitely many solutions. once you row reduce you get \[\begin{bmatrix}1&?\\0&0\end{bmatrix}\], so if y=t, then x=-?t. so any value of y gives a solution.

Answer 16

*any value of t

Answer 17

This means that there is a free variable. That free variable entails infinite solutions, since it will be in \(\mathbb R\).

Answer 18

They are linearly dependent