Real-Life Situation
Imagine you are organizing a birthday party and you decide to buy goodie bags for the children. Each goodie bag costs $8, and you want to include 9 additional candies in each bag. If your total budget for the goodie bags and candies is $144, how many goodie bags can you buy?
Equation
The total cost of the goodie bags can be represented by the equation: \[ 8(x + 9) = 144 \] where \( x \) is the number of goodie bags.
Solving the Equation
Method 1: Distributing and Isolating
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Distribute \( 8 \) on the left-hand side: \[ 8x + 72 = 144 \]
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Subtract \( 72 \) from both sides: \[ 8x = 144 - 72 \] \[ 8x = 72 \]
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Divide both sides by \( 8 \): \[ x = \frac{72}{8} \] \[ x = 9 \]
Thus, you can buy 9 goodie bags.
Method 2: Dividing First
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Divide both sides by \( 8 \): \[ x + 9 = \frac{144}{8} \] \[ x + 9 = 18 \]
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Subtract \( 9 \) from both sides: \[ x = 18 - 9 \] \[ x = 9 \]
Thus, you can also conclude that you can buy 9 goodie bags.
Conclusion
Both methods led us to the same solution, showcasing the power of algebraic manipulation.
Preferred Method
I prefer Method 2 (Dividing First) because it allows for a quicker simplification without needing to expand the equation first. This approach is often clearer and reduces the amount of arithmetic needed, making it easier to follow the steps to the solution. Additionally, dividing early helps to keep the numbers smaller and more manageable, which can reduce the likelihood of making mistakes during calculations.