Question

How do you solve the system of equations: 2x + y = 1 and 4x + 2y = −1?

Answer 1

`{(x = O/), (y = O/) :}`

Explanation:

Start by writing the system as given to you

`{(2x + y = 1), (4x + 2y = -1) :}`

Notice that you can simplify the second equation by dividing all the terms by `2` to get

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

This is equivalent to

`2x + y = -1/2`

Notice that the left side of the second equation is identical to the left side of the first equation, but that this expression equals two different values, `1` and `-1/2`, respectively.

In other words, you have

`{(2x + y = 1), (2x + y = -1/2) :}`

Let's say that you wanted to solve this system by substitution

`y = 1 -2 x`

`2x + (1 - 2x) = -1/2`

`color(blue)(cancel(color(black)(2x))) + 1 - color(blue)(cancel(color(black)(2x))) = -1/2`

`1 color(red)(!=) -1/2`

Since `1!=-1/2` for any value of `x` and of `y`, the system of equations has no real solution. You're essentially dealing with two parralel lines that will never intersect to produce a valid solution.