Let’s think about how many solutions an equation can have.
An equation is a statement that two expressions have an equal value. The equation
2x = 6
is a true statement if x is 3: 2 · 3 = 6.
It is a false statement if x is 4: 2 · 4 ≠ 6.
The equation 2x = 6 has one and only one solution, because there is only one number (3) that you can double to get 6.
Some equations are true no matter what the value of the variable is. For example:
2x = x + x
is always true, because if you double a number, that will always be the same as adding the number to itself. Equations like 2x = x + x have an infinite number of solutions. We say it is true for all values of x.
Some equations have no solutions. For example:
x = x + 1
has no solutions, because no matter what the value of x is, it can’t equal one more than itself. When we solve an equation, we are looking for the values of the variable that make the equation true. When we try to solve the equation, we make allowable moves assuming it has a solution. Sometimes we make allowable moves and get an equation like this:
8 = 7
This statement is false, so it must be that the original equation had no solution at all.
Which one doesn’t belong?
Without solving, identify whether these equations have a solution that is positive, negative, or zero.
Consecutive numbers follow one right after the other. An example of three consecutive numbers is 17, 18, and 19. Another example is -100, -99, -98.
How many sets of two or more consecutive positive integers can be added to obtain a sum of 100?
Are there sets with two numbers?
We can try two numbers grouped around 100/2 = 50.
49 + 50 ≠ 100 and 50 + 51 ≠ 100
And so, we cannot obtain two consecutive positive integers that will add to 100.
Are there sets with three numbers?
We can try three numbers grouped around 100/3 = 33.
Again, we cannot obtain three consecutive positive integers that will add to 100.
Similarly, we can try with four numbers.
We can try four numbers grouped around 100/4 = 25.
Again, we cannot obtain four consecutive positive integers that will add to 100.
We can try with five numbers.
We can try five numbers grouped around 100/5 = 20.
Yes, 18 + 19 + 20 + 21 + 22 = 100.
Eight? Yes, 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100.
Nine? No. ….
We can stop at 14, where the numbers would be grouped around 7.
At 15, the numbers would be grouped around 6 and the smallest number would be non-positive.
So there are two sets of consecutive positive integers whose sum is 100: one of five numbers (18 + 19 + 20 + 21 + 22), the other of eight (9 + 10 + 11 + 12 + 13 + 14 + 15 + 16).
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