Select the correct answer.
What is the justification for step 2 in the solution process?
[tex]\frac{1}{2} r+\frac{1}{2}=-\frac{2}{7} r+\frac{6}{7}-5[/tex]
Step 1: [tex]\frac{1}{2} r+\frac{1}{2}=-\frac{2}{7} r-\frac{29}{7}[/tex]
Step 2: [tex]\frac{1}{2} r=-\frac{2}{7} r-\frac{65}{14}[/tex]
A. the addition property of equality
B. the subtraction property of equality
C. the multiplication property of equality
D. the division property of equality
Answer (1)
Step 1 simplifies the original equation to 2 1 r + 2 1 = − 7 2 r − 7 29 .
Step 2 removes 2 1 from the left side, resulting in 2 1 r = − 7 2 r − 14 65 .
Subtracting 2 1 from both sides of Step 1 yields Step 2, as − 7 29 − 2 1 = − 14 65 .
The justification is the subtraction property of equality, so the answer is B.
Explanation
Understanding the Problem We are given the equation 2 1 r + 2 1 = − 7 2 r + 7 6 − 5 and two steps in its solution. Our goal is to determine the justification for the transition from Step 1 to Step 2.
Simplifying Step 1 Step 1 is 2 1 r + 2 1 = − 7 2 r − 7 29 . This step simplifies the right side of the original equation: 7 6 − 5 = 7 6 − 7 35 = − 7 29 .
Analyzing the Transition Step 2 is 2 1 r = − 7 2 r − 14 65 . Comparing Step 1 and Step 2, we observe that the term 2 1 has been removed from the left side of the equation. To achieve this, we must subtract 2 1 from both sides of Step 1.
Verifying the Subtraction Let's verify that subtracting 2 1 from the right side of Step 1 results in the right side of Step 2: − 7 29 − 2 1 = − 14 58 − 14 7 = − 14 65 . This confirms that we subtracted 2 1 from both sides.
Conclusion Since we subtracted the same value from both sides of the equation, the justification for this step is the subtraction property of equality.
Examples
The subtraction property of equality is a fundamental concept in algebra. It's used in various real-life scenarios, such as balancing budgets. For example, if you have a certain amount of money and you spend some, you subtract that amount from your initial balance to find out how much you have left. This principle ensures that the equation (your financial balance) remains true as long as you subtract the same amount from both sides (initial balance and expenses).
