Solving Chip Game Puzzle How Many Chips Did Alejandro Win Initially
In a game, Alejandro won a certain number of chips. Then he lost 3, after that he won the same number of chips as before, and lost 7 more chips. Finally, he was left with 26 chips. How many chips did Alejandro win the first time?
In this intricate puzzle, we delve into the world of Alejandro's chip game, carefully piecing together his wins and losses to unveil the initial number of chips he won. To solve this problem effectively, we'll embark on a step-by-step journey, meticulously tracking Alejandro's chip count as he navigates through the game. This meticulous approach will allow us to construct an algebraic equation, which will serve as our key to unlocking the solution. By carefully analyzing each step, we'll be able to isolate the unknown variable – the initial number of chips Alejandro won – and ultimately determine its value.
Setting Up the Equation: A Mathematical Representation of the Game
Let's represent the initial number of chips Alejandro won as "x." This variable will be the cornerstone of our equation, allowing us to translate the game's events into mathematical expressions. As we progress through the game's narrative, we'll meticulously track how Alejandro's chip count changes, incorporating each win and loss into our equation. This systematic approach will ensure that we capture all the essential elements of the game, setting the stage for a precise and accurate solution.
Step 1: Initial Win
At the beginning of the game, Alejandro wins a certain number of chips, which we've represented as "x." This initial win establishes Alejandro's starting chip count and serves as the foundation for our equation. To reflect this initial win in our equation, we simply add "x" to represent the chips Alejandro gains.
Step 2: First Loss
Next, Alejandro experiences his first setback, losing 3 chips. This loss directly impacts his chip count, reducing it by 3. To incorporate this loss into our equation, we subtract 3 from the expression, effectively reflecting the decrease in Alejandro's chips.
Step 3: Second Win
Undeterred by his earlier loss, Alejandro bounces back with another win, gaining the same number of chips he won initially, which is "x." This win replenishes his chip count and adds to his overall progress in the game. To represent this second win in our equation, we add "x" again, signifying the additional chips Alejandro gains.
Step 4: Second Loss
However, Alejandro's winning streak is short-lived as he faces another loss, this time losing 7 chips. This second loss further reduces his chip count, bringing him closer to his final chip total. To account for this loss in our equation, we subtract 7, reflecting the decrease in Alejandro's chips.
Step 5: Final Chip Count
After all the wins and losses, Alejandro is left with a final chip count of 26. This final chip count serves as the culmination of all the previous events and provides us with the crucial information we need to solve for "x." We set the expression we've built so far equal to 26, establishing the equation that will guide us to the solution.
Constructing the Equation: A Mathematical Representation
By carefully considering each step in the game, we can construct the following equation:
x - 3 + x - 7 = 26
This equation encapsulates the entire game, representing each win and loss as mathematical operations. It serves as a powerful tool for solving the puzzle, allowing us to isolate "x" and determine the initial number of chips Alejandro won.
Solving for x: Unveiling the Initial Chip Count
Now that we have our equation, the next step is to solve for "x," which represents the initial number of chips Alejandro won. To do this, we'll employ algebraic techniques to isolate "x" on one side of the equation. This process involves combining like terms, performing inverse operations, and simplifying the equation until we arrive at the value of "x."
Step 1: Combine Like Terms
The first step in solving for "x" is to combine like terms on the left side of the equation. We have two "x" terms, which we can combine to get 2x. Similarly, we can combine the constant terms, -3 and -7, to get -10. This simplification streamlines the equation, making it easier to work with.
Our equation now becomes:
2x - 10 = 26
Step 2: Isolate the Variable Term
Next, we want to isolate the term containing "x" on one side of the equation. To do this, we'll add 10 to both sides of the equation. This operation cancels out the -10 on the left side, leaving us with the variable term by itself.
Adding 10 to both sides, we get:
2x = 36
Step 3: Solve for x
Finally, to solve for "x," we need to get it by itself. We can do this by dividing both sides of the equation by 2. This operation isolates "x," revealing its value.
Dividing both sides by 2, we get:
x = 18
Therefore, Alejandro initially won 18 chips.
Verifying the Solution: Ensuring Accuracy
To ensure the accuracy of our solution, it's always a good practice to verify it by plugging the value of "x" back into the original equation. This process helps us confirm that our solution satisfies the conditions of the problem and that we haven't made any errors in our calculations.
Substituting x = 18 into the Original Equation
Let's substitute x = 18 into the original equation:
18 - 3 + 18 - 7 = 26
Simplifying the Equation
Now, let's simplify the equation by performing the arithmetic operations:
15 + 18 - 7 = 26
33 - 7 = 26
26 = 26
As we can see, the equation holds true when x = 18. This confirms that our solution is correct and that Alejandro initially won 18 chips.
Conclusion: Alejandro's Initial Triumph
In conclusion, by carefully analyzing the steps in Alejandro's chip game and constructing an algebraic equation, we have successfully determined that Alejandro initially won 18 chips. This problem highlights the power of algebraic techniques in solving real-world puzzles and underscores the importance of meticulous step-by-step analysis. The initial amount Alejandro won in the chip game was 18 chips.
Key Question: How many chips did Alejandro win initially in the game?
Keywords: algebraic equation, chip game, initial win, losses, solving for x, verify solution