Select the correct answer.
Paula used these steps to solve an equation:
Step 1: $-4(x+8)-2 x=25$
Step 2: $-4 x-32-2 x=25$
Step 3: $-6 x-32=25$
Step 4: $-6 x=57$
Step 5: $x=-9 \frac{1}{2}$
Between which two steps did Paula use the division property of equality?
A. steps 1 and 2
B. steps 2 and 3
C. steps 3 and 4
D. steps 4 and 5
Answer (2)
Distributive property is used to expand − 4 ( x + 8 ) to − 4 x − 32 .
Like terms − 4 x and − 2 x are combined to get − 6 x .
Addition property of equality is used, adding 32 to both sides of the equation.
Division property of equality is used, dividing both sides of the equation by -6, so the answer is D .
Explanation
Analyzing the steps Let's analyze Paula's steps to identify where the division property of equality was applied. The division property of equality states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. We need to check each step to see where this property was used.
Step 1 to Step 2 Step 1: − 4 ( x + 8 ) − 2 x = 25 Step 2: − 4 x − 32 − 2 x = 25 . Here, the distributive property is used to expand − 4 ( x + 8 ) to − 4 x − 32 .
Step 2 to Step 3 Step 2: − 4 x − 32 − 2 x = 25 Step 3: − 6 x − 32 = 25 . Here, like terms − 4 x and − 2 x are combined to get − 6 x .
Step 3 to Step 4 Step 3: − 6 x − 32 = 25 Step 4: − 6 x = 57 . Here, the addition property of equality is used, adding 32 to both sides of the equation.
Step 4 to Step 5 Step 4: − 6 x = 57 Step 5: x = − 9 2 1 . Here, the division property of equality is used, dividing both sides of the equation by -6. Specifically, x = − 6 57 = − 2 19 = − 9 2 1 .
Conclusion Therefore, the division property of equality was used between Step 4 and Step 5.
Examples
The division property of equality is a fundamental concept in algebra and is used extensively in solving equations. For example, if you want to determine the cost of one apple when you know the total cost of a bag of apples, you would use division. If 5 apples cost $2.50, you divide both sides of the equation 5 x = 2.50 by 5 to find the cost of one apple, which is $0.50. This principle applies to various real-world scenarios involving proportional relationships and resource allocation.
Paula used the division property of equality between Step 4 and Step 5 when she divided both sides of the equation by -6. This step is crucial in isolating the variable x. Thus, the correct answer is option D.
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