Select the correct answer. What is the justification for step 2 in the solution process? [tex]$12-x=7 x+32$[/tex] Step 1: [tex]$-x=7 x+20$[/tex] Step 2: [tex]$-8 x=20$[/tex] A. the addition property of equality B. the multiplication property of equality C. the division property of equality D. the subtraction property of equality
Answer (2)
To get from Step 1 ( − x = 7 x + 20 ) to Step 2 ( − 8 x = 20 ), we subtract 7 x from both sides of the equation.
This operation maintains the equality, as we are performing the same operation on both sides.
The subtraction property of equality states that if you subtract the same value from both sides of an equation, the equation remains balanced.
Therefore, the justification for Step 2 is the subtraction property of equality: D .
Explanation
Problem Analysis We are given the equation 12 − x = 7 x + 32 and the following steps:
Step 1: − x = 7 x + 20 Step 2: − 8 x = 20
We need to determine the justification for Step 2.
Justification for Step 2 To get from Step 1 to Step 2, we need to analyze the change that occurred. Step 1 is − x = 7 x + 20 , and Step 2 is − 8 x = 20 . We can see that 7 x was subtracted from both sides of the equation in Step 1.
Subtracting 7 x from both sides of − x = 7 x + 20 gives − x − 7 x = 7 x + 20 − 7 x , which simplifies to − 8 x = 20 .
Conclusion Therefore, the justification for Step 2 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 or calculating discounts. 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. Similarly, if you know the final price of an item after a discount and the amount of the discount, you can use subtraction to find the original price. These applications highlight the importance of understanding and applying the subtraction property of equality in everyday problem-solving.
The justification for Step 2 in the equation is the subtraction property of equality, which allows us to subtract the same amount from both sides without changing the equality. Therefore, the answer is D. This ensures that when we subtract 7 x from both sides, we maintain the balance of the equation.
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