Select the correct answer.
What is the justification for step 3 in the solution process?
[tex]0.8 a-0.1 a=a-2.5[/tex]
Step 1: [tex]0.7 a=a-2.5[/tex]
Step 2: [tex]-0.3 a=-2.5[/tex]
Step 3: [tex]a=8 . \overline{3}[/tex]
A. combining like terms
B. the subtraction property of equality
C. the addition property of equality
D. the division property of equality
Answer (2)
Start with the equation − 0.3 a = − 2.5 .
Divide both sides of the equation by − 0.3 to isolate a .
This operation is justified by the division property of equality.
The solution is a = 8. 3 .
Explanation
Problem Analysis We are given the equation 0.8 a − 0.1 a = a − 2.5 and the following steps:
Step 1: 0.7 a = a − 2.5 Step 2: − 0.3 a = − 2.5 Step 3: a = 8. 3
We are asked to identify the justification for Step 3.
Isolating the Variable To get from Step 2 to Step 3, we need to isolate a in the equation − 0.3 a = − 2.5 . We can do this by dividing both sides of the equation by − 0.3 .
Applying Division Property of Equality Dividing both sides of the equation − 0.3 a = − 2.5 by − 0.3 gives us:
− 0.3 − 0.3 a = − 0.3 − 2.5 a = 0.3 2.5 = 3 25 = 8. 3
The justification for this step is the division property of equality, which states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced.
Conclusion Therefore, the correct answer is D. the division property of equality.
Examples
The division property of equality is a fundamental concept in algebra that allows us to solve for unknown variables in equations. For example, if you know that 3 times a certain number is equal to 15, you can use the division property of equality to find that number by dividing both sides of the equation by 3. This gives you the equation x = 5, where x is the unknown number. This concept is used in various real-life scenarios, such as calculating the unit price of an item, determining the rate of speed, or solving for the dimensions of a geometric shape.
In Step 3 of the solution process, we isolate a from the equation − 0.3 a = − 2.5 by dividing both sides by − 0.3 . This operation is valid under the division property of equality, which ensures the balance of the equation when both sides are divided by the same non-zero number. Therefore, the correct answer is D. the division property of equality.
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