Real-Life Situation
Imagine you have a small bakery, and you are preparing to make custom gift boxes for a local event. Each gift box includes a certain number of cookies, and you want to pack them in a way that makes sure you have enough cookies for the event. You decide on adding 9 extra cookies to an initial amount of cookies you have, and to ensure you have enough, you decide to prepare 8 of these gift boxes. The total number of cookies you plan on preparing is 144 cookies.
We can express this situation with the equation:
\[ 8(x + 9) = 144 \]
Here, \(x\) represents the number of cookies you already have.
Solving the Equation
Let's solve the equation \( 8(x + 9) = 144 \) in two different ways.
Method 1: Distributive Property
-
Start with the original equation: \[ 8(x + 9) = 144 \]
-
Apply the distributive property: \[ 8x + 72 = 144 \]
-
Subtract 72 from both sides: \[ 8x = 144 - 72 \] \[ 8x = 72 \]
-
Divide both sides by 8: \[ x = \frac{72}{8} \] \[ x = 9 \]
So, you initially had 9 cookies.
Method 2: Dividing First
-
Start with the original equation: \[ 8(x + 9) = 144 \]
-
Divide both sides of the equation by 8: \[ x + 9 = \frac{144}{8} \] \[ x + 9 = 18 \]
-
Subtract 9 from both sides: \[ x = 18 - 9 \] \[ x = 9 \]
Again, we find that you initially had 9 cookies.
Conclusion
Both methods lead to the same solution of \(x = 9\).
Preference for Solution Method
I prefer Method 2, where we divided first, because it simplifies the equation faster and allows you to eliminate the coefficient of \(8\) right away. This reduces the complexity, and you avoid combining steps, making it less likely to make a mistake in arithmetic. The division clearly shows the relationship between \(x\) and the total cookies needed, and I find this method to be more straightforward and intuitive.