- The objective is to identify the first step for solving the equation \(5 + 3(4x + 8) = 9x + 12 - 6x\).
- Using the distributive property, we expand \(3(4x + 8)\) to \(12x + 24\), which removes the parentheses and is the standard first step in algebraic simplification.
- Other options fail because they attempt to combine terms before multiplication, violating the order of operations.
- The correct first step is: Makayla could multiply \(4x\) and \(8\) by \(3\) to remove the parentheses, which leads to the final answer: \(\boxed{\text{Makayla could multiply 4x and 8 by 3 to remove the parentheses}}\).
Explanation
- Analysis of the Equation
The given equation is \(5 + 3(4x + 8) = 9x + 12 - 6x\). To solve this linear equation efficiently, we must follow the standard order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). The presence of parentheses on the left side of the equation dictates that our first focus should be to clear these parentheses using the distributive property. This allows us to combine like terms and eventually isolate the variable \(x\).
- Evaluating Potential First Steps
Let's evaluate each suggested step:
- First option (combining 5, 3, and 8): This is incorrect because it violates the order of operations. The 3 is a coefficient multiplying the entire expression \((4x + 8)\), so it must be distributed before performing addition with the 5.
- Second option (subtracting 4x): This is not a valid first step because the \(4x\) is inside the parentheses, making it part of a multiplication operation with the 3. We cannot isolate it until the multiplication is resolved.
- Third option (adding 5 and 3): This is also incorrect. Similar to the first option, the 3 is a multiplier, not an addend for the 5.
- Fourth option (distributing 3): This is the correct step. Applying the distributive property \(a(b + c) = ab + ac\) allows us to write \(3(4x + 8)\) as \(3(4x) + 3(8) = 12x + 24\). This effectively removes the parentheses and leaves us with a simplified expression on the left side: \(5 + 12x + 24\), which can then be easily simplified further.
- Justification of the Correct Step and Conclusion
By applying the distributive property, we transform the equation into a form that is much easier to manage. Distributing the 3 correctly turns \(5 + 3(4x + 8)\) into \(5 + 12x + 24\). Combining like terms then gives \(12x + 29\). On the right side, \(9x - 6x + 12\) simplifies to \(3x + 12\). The resulting equation is \(12x + 29 = 3x + 12\), which is a standard linear equation ready for further simplification. Therefore, the correct first step is to multiply \(4x\) and \(8\) by \(3\).
Examples
Imagine you are managing a budget for three separate school clubs, and each club needs a set fee of 4 dollars per student plus a fixed 8 dollar equipment cost. If you have 5 dollars left in your general fund, your total spending can be expressed as 5 plus 3 times the sum of the fee and the equipment. Understanding how to correctly distribute this multiplier ensures you accurately calculate your total expenses before comparing them to your available funds. This simple algebraic process is fundamental to balancing any budget or resource allocation problem where costs are grouped together.