Question

How do you solve the system `y = -x + 2`, `2y = 4 - 2x`?

Answer 1

`(x,-x+2)`

Explanation:

`y=-x+2color(white)(88)[1]`

`2y=4-2xcolor(white)(888)[2]`

We can solve this one by substitution. Notice we already know the value of `y` in terms of `x` from equation `[1]`. Substituting this in equation `[2]`, give us:

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

`-2x+4=4-2x`

`0=0`

This is known as linear dependence. What this means is we have two equations, but one is just a multiple of the other. Notice:

`2y=4-2x`

Is just `bb(y=-x+2)` multiplied by `bb2`:

`2(y=-x+2)`

In this situation we have to assign an arbitrary value to one of the variables and express the other variable in terms of this. So for arbitrary `x`

`y=-x+2`

We can write the solutions as:

`(x,y)->(x,-x+2)`

Since we are assigning a value to `x` and calculating the corresponding value of `y`, this gives us an infinite number of solutions.

If these two equations are graphed we find that they are exactly the same line.