Question

How do you solve the following linear system: # -x+2y=-6 , 4x+3y=0 #?

Answer 1

`x=18/11`
Solution method to find x is shown in detail. I will let you solve for y by substituting for `x` in Eqn `(1_a) " or "(2_a)`
The methods I used to find `x` help with all sorts of Algebra.

Explanation:

Given:
`-x+2y=-6 ..................................(1)`
`4x+3y=0........................................(2)`

rearrange the equations to give:

`y=1/2x-3...................................(1_a)`
`y=-4/3x.....................................(2_a)`

Equating equation `color(brown)((1_a))` to equation `color(brown)((2_a))` through `color(blue)(y)`

`color(brown)(1/2x-3 =) color(blue)(y) color(brown)(= -4/3x )`

`1/2x-3=-4/3x`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add `color(blue)(4/3x)` to both sides:

`color(brown)((1/2x-3)) color(blue)(+4/3x) = color(brown)((-4/3x)) color(blue)(+4/3x)`

`1/2x+4/3x -3 =0`

`(3+8)/6x -3=0`

`11/6 x -3 =0`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add `color(blue)(3)` to both sides

`color(brown)((11/6x-3)) color(blue)(+3) =color(brown)((0))color(blue)(+3)`

`11/6x +0 = 3`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Multiply both sides by `color(blue)(6/11)`

`color(brown)((11/6x) )color(blue)(times 6/11)=color(brown)((3))color(blue)(times 6/11)`

`color(green)(x=18/11)`