alternative to factoring

rkav wrote:*plugging in x into original equation*
...
Everything cancelled out. Where did I go wrong? Thanks!

Everything cancels out means the plugged value satisfy the equation.
It is obvious that if you get the value of x from an equation and then plugged the same value for x in the same equation, the value will always satisfy the equation, i.e. everything will cancel out.

rkav wrote:A fruit stand sold apples for .70 each and bananas for .50 each. If a customer purchased both apples and bananas from the stand for a total of 6.30 what total number of bananas and apples did the customer purchase?

Let us assume the customer purchased x apples and y bananas, where x and y are positive integers.
Hence, (0.7)*x + (0.5)*y = 6.30

So, 7x + 5y = 63
--> x = (63 - 5y)/7

As x must be a positive integer, (63 - 5y) must be a multiple of 7.
As 63 is a multiple of 7, 5y must be a multiple of 7 ---> y must be a multiple of 7.

Take y = 7 ---> x = (63 - 5*7)/7 = 4
Take y = 14 ---> x = (63 - 5*14) < 0

Only possible value of y is 7.
Hence, x = 4 and y = 7.