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

Question

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

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Answer 1 · 2018-02-24T13:49:31

See a solution process below:

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]}