Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given 9x - 3y = 3 and -3x - 8y = 17?

Answer 1

`x=1`
`y=-2`

Explanation:

`9x-3y=3` ---------------(1)
`-3x-8y=17`--------------(2)

For two linear equations to have a solution, their slopes must be different.
In that case, they will have one solution.
If both are parallel (Two slopes are equal), the equations will have not solution.

If one equation is a multiple of another equation, they will have many solutions.

To have a solution let us multiply equation (1) with `3` and multiply equation (2) with `9`.

`9x-3y=3---------(1) xx 3`
`-3x-8y=17 -------(2) xx 9`

`27x-9y=9` ---------------(1) ------Add both the equations.
`-27x-72y=153`--------------(2)

`-81y=162`
`y=162/(-81)=-2`

`y=-2`

Substitute >`y=-2` in equation (1)

`9x-3(-2)=3`
`9x-6=3`
`9x=3+6=9`
`x=9/9=1`
`x=1`