To create an entry code, you must first choose 2 letters and then, 4 single-digit numbers. How many different entry codes can
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Start by writing the system down, I will use to represent
Substitute the fact that into the first equation to get,
Simplify into a quadratic form (),
Now you can use Vieta's rule which states that any quadratic equation can be written in the following form,
which then must factor into
And the solutions will be .
Clearly for small coefficients like ours , this is very easy to figure out. To get 5 and 6 we simply say that
.
This fits the definition as and
.
So as mentioned, solutions will equal to but these are just x-values in the solution pairs of a form
.
To get y-values we must substitute 3 for x in the original equation and then also 2 for x in the original equation. Luckily we already know that substituting either of the two numbers yields a zero.
So the solution pairs are and
.
Hope this helps :)
