i NEED MORE THAN JUST THE EXAMPLE i ALSO NEED TO EXPLANATION
Take any number (except for 1). Square that number and then subtract one. Divide by one less than your original number. Now subtract your original number. You reached 1 for an answer, didn’t you? How does this number game work? (Hint: Redo the number game using a variable instead of an actual number and rewrite the problem as one rational expression). How did the number game use the skill of simplifying rational expressions? Create your own number game using the rules of algebra and post it for your classmates to solve. Be sure to think about values that may not work. State whether your number game uses the skill of simplifying rational expressions
Take any number (except for 1) --- x
Square that number and then subtract one --- x^2 - 1
Divide by one less than your original number --- (x^2 - 1)/(x-1)
= (x+1)(x-1)/(x-1)
= x+1
Now subtract your original number --- x+1 - x
= x
To understand how this number game works, let's break it down step by step using a variable instead of an actual number.
Step 1: Take any number (let's call it x).
Step 2: Square that number (x^2).
Step 3: Subtract 1 from the squared number (x^2 - 1).
Step 4: Divide by one less than your original number (x^2 - 1) / (x - 1).
Step 5: Subtract your original number (x^2 - 1) / (x - 1) - x.
The goal is to show that, regardless of the value of x (except for 1), the expression eventually simplifies to 1.
To demonstrate this, let's simplify the expression:
Step 6: Multiply the expression by the conjugate of the denominator to remove the denominator and simplify the expression:
[(x^2 - 1) / (x - 1) - x] * [(x - 1) / (x - 1)] = (x^2 - 1 - x(x - 1)) / (x - 1)
Simplifying this gives: (x^2 - 1 - x^2 + x) / (x - 1) = (x - 1) / (x - 1)
Step 7: Cancel the common factor of (x - 1) in the numerator and denominator:
(x - 1) / (x - 1) = 1
Therefore, no matter what value of x you choose (except for 1), the expression simplifies to 1. This is why you always reach 1 as the answer in the number game.
Now, let's create our own number game using the rules of algebra and see if it involves simplifying rational expressions.
For example, let's say our number game involves the following steps:
Step 1: Take any number (let's call it y).
Step 2: Multiply that number by 3 (3y).
Step 3: Add 5 to the resulting number (3y + 5).
Step 4: Divide by 2 (3y + 5) / 2.
Step 5: Subtract your original number (3y + 5) / 2 - y.
To solve this game, we can simplify the expression:
Step 6: Multiply the expression by the common denominator 2 to eliminate the fraction:
[(3y + 5) / 2 - y] * (2/2) = (3y + 5 - 2y) / 2 = (y + 5) / 2
In this number game, we did not encounter rational expressions. The expression (y + 5) / 2 is already simplified, so the game does not involve the skill of simplifying rational expressions.
In conclusion, the original number game involved simplifying a rational expression by canceling out a common factor. Our created number game, on the other hand, did not involve simplifying rational expressions.