How do you solve the following system?: `4x +5y =-4 , -3x -8y = 4`

Multiplying the first equation by `3` and the second one by `4` and adding both we get
`-17y=4`
so

`y=-4/17`
now we can calculate `x`
`4x+5(-4/17)=-4`
so

`x=1/4*(-4+20/17)`

so

`x=-12/17`

Given:

`4x+5y=-4" "...........................Equation(1)`
`-3x-8y=4" "............................Equation(2)`

2 equations and 2 unknowns. Thus solvable.

We need to end up with one equation and 1 unknown.

`color(blue)("'Getting rid' of the "x" term")`

`[color(white)("d")3xxEqn(1)color(white)(2/2)]+[color(white)("d")4xxEqn(2)color(white)(2/2)]`

`color(white)("d.")12x+15y=-12" "........................Equation(1_a)`
`ul(-12x-32y=color(white)("d..")16)" "........................Equation(2_a)`
`color(white)("dd")0color(white)(".d")-17y=+4`

Divide both sides by `-17`

`color(white)("dddddddd")y=-4/17`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("'Getting rid' of the "y" term")`

Substitute `-4/17` for `y`

I choose `Eqn(1)` as all the left side of = is positive.
It will still work if you choose `Eqn(2)`

`color(green)( 4x+5color(red)(y)=-4 color(white)("dddd")->color(white)("dddd")4x+5(color(red)(-4/17))=-4 )`

`color(white)("dddddddddddddddd")->color(white)("dddd")4xcolor(white)("ddd")-20/17color(white)("dd")=-4`

Add `20/17` to both sides

`color(white)("dddddddddddddddd")->color(white)("dddd")4xcolor(white)("dd")=-4+20/17`

`color(white)("dddddddddddddddd")->color(white)("dddd")4x=-48/17`

Divide both sides by 4

`color(white)("ddddddddddddddd")->color(white)("dddd")x=-12/17`