How do you solve `y=|x|` and `y=x+2` using substitution?

Use the substitution `y = absx`

`y = x+2`

`absx = x + 2`

Now subtract `x` from both sides.

`absx - x = 2`

Hmm... how do we simplify this? Well, we have three cases:

1. `" "x` is positive
2. `" "x` is zero
3. `" "x` is negative

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If `x` is positive, then `absx = x`, so we can use this substitution:

`absx - x = 2, " " x >0`
`x-x =2, " "color(white). x>0`
`0 = 2, " "" "" "x>0`

And since `0` is never equal to `2`, this means we have no positive solutions.

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Next, let's try case 2, when `x=0`.

`|0| - 0 = 2`

`0 = 2`

Again, `0` cannot equal `2`, so our solution cannot be `x=0`.

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Finally, when `x` is negative, `absx` is the same as `-x`, since to make a negative number positive, you add another negative sign. (Two negatives make a positive!) So, we can use the substitution `absx = -x`:

`absx - x = 2, " " x<0`
`-x-x = 2, " " x<0`
`-2x = 2, " "" " x<0`

Now we can divide both sides by `-2` to find `x`:

`(-2x)/(-2) = 2/(-2), " "x<0`

`x = -1`

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So this is the `x` coordinate for our solution. To find the `y` coordinate, we simply plug `x=-1` back into one of the original equations and solve for `y`.

`y = absx`
`y = abs(-1`
`y = 1`

So `x=-1` and `y=1`. Therefore, our solution is `(-1,1)`.

Final Answer