How many ordered pairs of solutions (a,b) are there to:
a-b=2
where, a and b lie between [-100,100]
On a table there is a bowl of chocolates. All but four are yellow. All but four are blue. All but four are purple. How many chocolates are in the bowl?
To determine the number of ordered pairs of solutions (a, b) that satisfy the equation a - b = 2 within the given range [-100, 100], we need to analyze the possibilities.
We will start by expressing one variable in terms of the other to help us generate possible values within the range.
Rearrange the equation a - b = 2:
a = b + 2.
Now, let's consider the possible values for the variable b. Since b can range from -100 to 100 (inclusive), there are 201 possible values for b within this range.
For each value of b, we can calculate the corresponding value of a using the equation a = b + 2.
Let's walk through an example:
When b = -100, a = (-100) + 2 = -98.
When b = -99, a = (-99) + 2 = -97.
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When b = 0, a = 0 + 2 = 2.
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When b = 99, a = 99 + 2 = 101.
When b = 100, a = 100 + 2 = 102.
As we can see, the values of a fall outside of the range [-100, 100] for some values of b. Therefore, the only valid ordered pairs (a, b) that satisfy the equation a - b = 2 within the given range are:
(-98, -100), (-97, -99), ..., (2, 0), (-101, 99), (-102, 100).
Counting these pairs, we find that there are 201 ordered pairs of solutions (a, b) that satisfy the equation a - b = 2 within the range [-100, 100].