Question

Show that the pair of linear equations x=2y and y=2x has an uniqure solution at (0,0). How to solve this?

Show that the pair of linear equations x=2y and y=2x has an unique solution at (0,0). Justify your answer.

Answer 1

See a solution process below:

Explanation:

Step 1) Because the first equation is already solved for `x` we can substitute `2y` for `x` in the second equation and solve for `y`:

`y = 2x` becomes:

`y = 2 * 2y`

`y = 4y`

`y - color(red)(y) = 4y - color(red)(y)`

`0 = 4y - 1color(red)(y)`

`0 = (4 - 1)color(red)(y)`

`0 = 3y`

`0/color(red)(3) = (3y)/color(red)(3)`

`0 = (color(red)(cancel(color(black)(3)))y)/cancel(color(red)(3))`

`0 = y`

`y = 0`

Step 2) We can now substitute `0` for `y` in the first equation and calculate `x`:

`x = 2y` becomes:

`x = 2 * 0`

`x = 0`

Therefore the solution is:

`x = 0` and `y = 0`

Or

`(0, 0)`

We can also graph these equations showing the solution:

graph{(x-2y)(y-2x) = 0 [-5, 5, -2.5, 2.5]}