- The student begins by applying the Distributive Property to simplify the expression \(0.5(8x - 6)\) into \(4x - 3\).
- Following this, the student uses the Addition Property of Equality to transform \(4x - 3 = 9\) into \(4x = 12\).
- Finally, the student uses the Division Property of Equality to solve for \(x\), arriving at \(x = 3\).
- Substituting \(x = 3\) back into the original equation confirms that the result is correct and the solution is valid.
- The first property used is the Distributive Property, and the final solution is \(\boxed{x=3}\).
Explanation
- Analysis of the First Step
The problem provides a step-by-step solution to the linear equation \(0.5(8x - 6) = 9\). To identify the first property used, we examine the transformation from the original equation to the expression in Step 1. The original equation is \(0.5(8x - 6) = 9\). In Step 1, the student writes \(4x - 3 = 9\). By distributing the \(0.5\) to the terms inside the parentheses, we get \(0.5 \times 8x = 4x\) and \(0.5 \times (-6) = -3\). This transformation, \(a(b + c) = ab + ac\), is defined as the Distributive Property. Therefore, the first property used by the student is the Distributive Property.
- Verification of Remaining Steps and Conclusion
Next, we verify the remaining steps. Step 1 is \(4x - 3 = 9\). Step 2 is \(4x = 12\). This was achieved by adding \(3\) to both sides of the equation, which is the Addition Property of Equality. Step 3 is \(x = 3\), which was achieved by dividing both sides of the equation by \(4\), which is the Division Property of Equality. To confirm the validity of the final solution, we substitute \(x = 3\) into the original equation: \(0.5(8(3) - 6) = 0.5(24 - 6) = 0.5(18) = 9\). Since the substitution results in \(9 = 9\), the final solution is correct. Thus, the first property used is the Distributive Property and the solution is valid.
Examples
The Distributive Property is an essential tool in algebra used to simplify expressions. For example, if you are calculating the total cost of 5 notebooks priced at \(\$x\) each and 5 pens priced at \(\$2\) each, you can express the cost as \(5(x + 2)\) or as \(5x + 10\). Being able to distribute the multiplier across the terms inside the parentheses allows you to solve for unknowns more efficiently in various real-world budgeting and planning scenarios.