Thread: 6x-y = 21 and 6y - x = 14, x-y?

1. 6x-y = 21 and 6y - x = 14, x-y?

As the title says, I want to find x-y, knowing that:

6x-y = 21 and 6y - x = 14

I have used substition methods but it seems to be complicating the question which looks as if it should be solved quickly because of its context.

Any help of how to tackle this problem would be appreciated.

2. Originally Posted by david18
As the title says, I want to find x-y, knowing that:

6x-y = 21 and 6y - x = 14

I have used substition methods but it seems to be complicating the question which looks as if it should be solved quickly because of its context.

Any help of how to tackle this problem would be appreciated.
subtract the second equation from the first. do you see it?

3. Try to cancel a variable:

$\displaystyle 6x - y = 21$
$\displaystyle 6y - x = 14$

So I'll try to cancel the x's here:
$\displaystyle 6(6y - x) = 6(14)$
$\displaystyle 36y - 6x = 84$

So now we can cancel the x's to get:

$\displaystyle 6x - y = 21 \quad+$
$\displaystyle 36y - 6x = 84$

$\displaystyle 35y = 105$
$\displaystyle y = 3$

Plug that back into the original equation and:
$\displaystyle 6x - 3 = 21$
$\displaystyle 6x = 24$
$\displaystyle x = 4$
Check the values if you want to to make sure they satisfy both equations.

4. Originally Posted by SnipedYou
Try to cancel a variable:

$\displaystyle 6x - y = 21$
$\displaystyle 6y - x = 14$

So I'll try to cancel the x's here:
$\displaystyle 6(6y - x) = 6(14)$
$\displaystyle 36y - 6x = 84$

So now we can cancel the x's to get:

$\displaystyle 35y = 105$
$\displaystyle y = 3$

Plug that back into the original equation and:
$\displaystyle 6x - 3 = 21$
$\displaystyle 6x = 24$
$\displaystyle x = 4$
Check the values if you want to to make sure they satisfy both equations.
this problem was meant to be quick. subtracting the second equation from the first is the fastest way. because you immediately get 7x - 7y = 7 => x - y = 1

so it's a two liner (and just because we want to be neat, it could be one line if we wish)

5. Hello, David!

Given: .$\displaystyle \begin{array}{ccc}6x-y &= & 21 \\ 6y - x &= & 14\end{array}$

$\displaystyle \text{Find }\,x-y$

We have: .$\displaystyle \begin{array}{ccc}6x-y & = & 21 \\ x - 6y & = & -14\end{array}$

Add the equations: .$\displaystyle 7x - 7y \:=\:7$

. . . . . Divide by 7: .$\displaystyle \boxed{x - y \:=\:1}$

Darn, too slow again!
.