Question

The system of equations below is consistents with infinity many solutions where a and b are constants.Find the values of a and b. 2x+y=6 ax+by=9

Answer 1

The values of
`a` and
`b` so that the two linear equations have infinite solutions are
`3` and
`(3)/(2)`, respectively.

Explanation:

Two linear equations with two variables have infinite solution if and only if they are linearly dependent. That is, one linear equation is a multiple of the other one. Let be the following system of linear equations:

`2\cdot x + y = 6` (1)

`a\cdot x + b\cdot y = 9` (2)

The following condition must be observed:

`r = (a)/(2) = (b)/(1) = (9)/(6)` (3)

After some quick operations, we find the following information:

`r = (3)/(2)`,
`a = 3`,
`b = (3)/(2)`

The values of
`a` and
`b` so that the two linear equations have infinite solutions are
`3` and
`(3)/(2)`, respectively.