How do solve the following linear system?: `4x-2y=2 , x-4y=4`?

Step 1) Solve the second equation for `x`:

`x - 4y = 4`

`x - 4y + color(red)(4y) = 4 + color(red)(4y)`

`x - 0 = 4 + 4y`

`x = 4 + 4y`

Step 2) Substitute `(4 + 4y)` for `x` in the first equation and solve for `y`:

`4x - 2y = 2` becomes:

`4(4 + 4y) - 2y = 2`

`(4 xx 4) + (4 xx 4y) - 2y = 2`

`16 + 16y - 2y = 2`

`16 + (16 - 2)y = 2`

`16 + 14y = 2`

`-color(red)(16) + 16 + 14y = -color(red)(16) + 2`

`0 + 14y = -14`

`14y = -14`

`(14y)/color(red)(14) = -14/color(red)(14)`

`(color(red)(cancel(color(black)(14)))y)/cancel(color(red)(14)) = -1`

`y = -1`

Step 3) Substitute `-1` for `y` in the solution to the second equation at the end of Step 1 and calculate `x`:

`x = 4 + 4y` becomes:

`x = 4 + (4 * -1)`

`x = 4 + (-4)`

`x = 4 - 4`

`x = 0`

The solution is: `x = 0` and `y = -1` or `(0, -1)`

`color(white)(-)4x-2y=2`
`color(white)(-)`
`color(white)(-)x-4y=4`

Let's try elimination! If we multiply the first equation by `2`, the `y`s will have equal coefficients, and we'll be able to eliminate them from the set:

`color(white)(-)2(4x-2y=2)`
`color(white)(-)`
`color(white)(-2..)x-4y=4`

`color(white(--)`

`color(white)(-)8x-4y=4`
`color(black)(-)`
`color(white)(-)x-4y=4`
...........................................
`7x=0`

divide by `7` on both sides

`x=0`

Now let's solve for `y`!

`x-4y=4`

substitute `0` for `x`

`(0)-4y=4`

`-4y=4`

divide by `-4` on both sides

`y=-1`

To check our work, let's substitute `x` and `y` for `0` and `-1` in the other equation.

`4x-2y=2`

`4(0)-2(-1)` should equal `2`, if we did our math right...

`0--2`

`2=2`
We were right!!!

`x=0, y=-1`

Just to triple check, let's graph the equations and see where they intercept:

They cross at `(0,-1)`, where `x=0` and `y=-1`