Question

How do you solve by substitution `x+-y=1` and `-2x+y=4`?

Answer 1

The problem has two solutions:
`x=-1,y=2`.
`x=-5,y=-6`.

Explanation:

To solve by substitution, we isolate one of the variables in one equation, and use that to solve the other equation.

`x+-y =1` shows two cases: `x+y=1` or `x-y=1`.

Case 1:
Our system is:

`x+y=1`
`-2x+y=4`

Since the first equation is easier to work with, we can isolate a variable there.
`x+y=1`
`x=1-y`

Now, we substitute `1-y` into `x` in the other equation, and simplify.

`-2x+y=4`
`-2(1-y)+y=4`
`-2+2y+y=4`
`3y=6`
`y=2`

We now know the value of `y`, which we can use to solve for `x` using either equation:

`x+y=1`
`x+2=1`
`x=-1`

Therefore, the solution to Case 1 is `x=-1,y=2`.

Case 2:
Our system is:

`x-y=1`
`-2x+y=4`

Like before, we isolate one of the variables in one equation and solve using the other equation.

`x-y=1`
`x=1+y`

`-2x+y=4`
`-2(1+y)+y=4`
`-2-2y+y=4`
`-y=6`
`y=-6`

At this point, we know the `y=-6`, so we can solve for `x` using either equation:

`x-y=1`
`x+6=1`
`x=-5`

Therefore the solution to case 2 is `x=-5,y=-6`.

The problem has two solutions:
`x=-1,y=2`.
`x=-5,y=-6`.