Select the correct answer.
What is the justification for step 2 in the solution process?
[tex]$12-x=7 x+32$
Step 1: [tex]$-x=7 x+20$
Step 2: [tex]$-8 x=20$
A. the division property of equality
B. the multiplication property of equality
C. the addition property of equality
D. the subtraction property of equality
Answer (2)
We are given the equation 12 − x = 7 x + 32 and the steps to solve it.
We analyze the transition from Step 1 ( − x = 7 x + 20 ) to Step 2 ( − 8 x = 20 ).
We determine that subtracting 7 x from both sides of the equation in Step 1 results in Step 2.
Therefore, the justification for Step 2 is the subtraction property of equality, so the answer is D .
Explanation
Understanding the Problem 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 find the justification for step 2.
Analyzing the Transition To get from Step 1 to Step 2, we need to perform an operation on both sides of the equation in Step 1 such that the equation remains balanced. Let's analyze the transition from Step 1 to Step 2.
Step 1: − x = 7 x + 20 Step 2: − 8 x = 20
We can see that the term 7 x has been eliminated from the right side of the equation. To eliminate 7 x , we need to subtract 7 x from both sides of the equation in Step 1:
− x − 7 x = 7 x + 20 − 7 x
This simplifies to:
− 8 x = 20
Justification for Step 2 Since we subtracted 7 x from both sides of the equation, the justification for Step 2 is the subtraction property of equality.
Final Answer The correct answer is D. the subtraction property of equality.
Examples
The subtraction property of equality is a fundamental concept in algebra and is used in various real-life scenarios. For example, if you have a balance scale with equal weights on both sides, and you remove the same weight from both sides, the scale will remain balanced. Similarly, in financial transactions, if you subtract the same amount from both your income and expenses, your net balance will still reflect the correct difference. This principle ensures fairness and accuracy in various mathematical and practical applications.
To justify Step 2 in the equation solving process − x = 7 x + 20 to − 8 x = 20 , we used the subtraction property of equality. This property allows us to subtract the same value from both sides of an equation while keeping it balanced. Thus, the chosen answer is D.
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