How do you solve the following system?: `4x + 5y = 2 , 17 x + 2y = 6`

There are a couple of ways we can solve this. I'm going to use the elimination method
`4x+5y=2`
`17x+2y=6`

I need to make two of the variables equal, so I'm going to multiply the second equation by `2.5`. That will change `2y` into `5y`.

`4x+5y=2`
`2.5(17x+2y=6)` or `42.5x+5y=15`

Now we subtract the two equations:
`4x+5y=2`
`-`
`42.5+5y=15`
`color(black)(------)`
`-38.5x+0y=-13`

If we simplify our equation, we find that `x=(-13)/-38.5` or `x=26/77`

Now we just solve for `y`. We can use either of the two equations. I like the first one (it has nicer numbers than `17`).

`4(color(purple)(x))+5y=2`
`4(color(purple)(26/77))+5y=2`
`cancel(104/77)+5y=2`
`cancel(-104/77)color(white)(+5y)-104/77`

Now we have
`5y=50/77`
or
`y=10/77`.

To double check our work, we need to plug our values into one (or both) of the equations.
`4(26/77)+5(10/77)` should equal `2`
`104/77+50/77`
`2=2`, so we were right!
Just to be safe, let's look at the other equation.

`17(26/77)+2(10/77)` should equal `6`
`442/77+20/77`
`6=6`
Good job, we got it right! Nice work

We can solve this system of equations using the elimination method. First, we can set up the system in such a way that the `y` terms cancel out. To do this, we need both the `y` terms in each equation to have the same coefficient (one negative and one positive):

`2(4x+5y=2)`
`-5(17x+2y=6)`

So we get:
`8x+10y=4`
`-85x-10y=-30`

Adding these two equations together gives us:
`-77x=-26`

`x=26/77=0.338`

Plug this value of `x` back into one of the given equations:
`4(0.338)+5y=2`

`y=(2-4(0.338))/5=0.130`

So, `x=0.338` and `y=0.130`