- Analyze the transition: To move from \(7x - 25 = 4x - 1\) to \(7x = 4x + 24\), we add \(25\) to both sides.
- Identify the property: Adding the same value to both sides of an equation is defined by the addition property of equality.
- State the final answer: The justification for Step 2 is the addition property of equality.
- Final Answer: \(\boxed{\text{The addition property of equality}}\)
Explanation
- Analyzing the Step-by-Step Transition
The problem asks for the justification for moving from the equation \(7x - 25 = 4x - 1\) (Step 1) to the equation \(7x = 4x + 24\) (Step 2). We observe that in the transition from Step 1 to Step 2, the constant term \(-25\) on the left side of the equation has been eliminated and a corresponding term \(+24\) has appeared on the right side. This transformation corresponds to adding \(25\) to both sides of the original equation in Step 1:
\[7x - 25 + 25 = 4x - 1 + 25\]
\[7x = 4x + 24\]
Since we added the same value to both sides of the equation to maintain equality, this process is justified by the addition property of equality.
- Justifying the Operation
The addition property of equality states that for any real numbers \(a, b,\) and \(c,\) if \(a = b,\) then \(a + c = b + c.\) In our case, \(a = 7x - 25\), \(b = 4x - 1\), and \(c = 25\). Adding 25 to both sides of the equality \(a = b\) yields \(7x - 25 + 25 = 4x - 1 + 25\), which simplifies to \(7x = 4x + 24\). Thus, the justification for step 2 is the addition property of equality.
Examples
In real life, the addition property of equality is used whenever you need to balance an equation, such as when calculating budget expenses. For example, if you know that the amount you spent on two items (item A and item B) is equal to your total budget minus your remaining cash, you can rearrange the equation to find the total cost of the items by adding the 'cash spent' back to both sides. This ensures the equality is maintained and helps you track your finances accurately.