Select the correct answer.
Henry used these steps to solve an equation:
Step 1: [tex]$-3(x+2)=5(x-7)$[/tex]
Step 2: [tex]$\quad-3 x-6=5 x-35$[/tex]
Step 3: [tex]$\quad-8 x-6=-35$[/tex]
Step 4 : [tex]$-8 x=-29$[/tex]
Step 5 : [tex]$\quad x=\frac{29}{8}$[/tex]
Between which two steps did Henry apply the distributive property?
A. steps 1 and 2
B. steps 2 and 3
C. steps 3 and 4
D. steps 4 and 5
Answer (2)
The distributive property is applied to expand expressions of the form a ( b + c ) into ab + a c .
In Step 1, the equation is − 3 ( x + 2 ) = 5 ( x − 7 ) .
In Step 2, the equation becomes − 3 x − 6 = 5 x − 35 , which shows the application of the distributive property on both sides of the equation.
Therefore, the distributive property was applied between steps 1 and 2. The answer is steps 1 and 2.
Explanation
Analyzing the steps Let's analyze the steps Henry took to solve the equation and identify where the distributive property was applied. The distributive property states that a ( b + c ) = ab + a c . We need to see which step involves expanding expressions in this way.
Identifying the Distributive Property Step 1: − 3 ( x + 2 ) = 5 ( x − 7 ) Step 2: − 3 x − 6 = 5 x − 35
Looking at the left side of the equation, we see that − 3 ( x + 2 ) becomes − 3 x − 6 . This is the distributive property: − 3 ∗ x + ( − 3 ) ∗ 2 = − 3 x − 6 .
On the right side, 5 ( x − 7 ) becomes 5 x − 35 . Again, this is the distributive property: 5 ∗ x + 5 ∗ ( − 7 ) = 5 x − 35 .
So, the distributive property was applied between Step 1 and Step 2.
Checking Step 3 Step 3: − 8 x − 6 = − 35 This step combines the 'x' terms from Step 2. To get from − 3 x − 6 = 5 x − 35 to − 8 x − 6 = − 35 , we subtract 5 x from both sides and get − 3 x − 5 x − 6 = 5 x − 5 x − 35 which simplifies to − 8 x − 6 = − 35 . This is not the distributive property.
Checking Step 4 Step 4: − 8 x = − 29 This step isolates the 'x' term. To get from − 8 x − 6 = − 35 to − 8 x = − 29 , we add 6 to both sides: − 8 x − 6 + 6 = − 35 + 6 , which simplifies to − 8 x = − 29 . This is not the distributive property.
Checking Step 5 Step 5: x = 8 29 This step solves for 'x'. To get from − 8 x = − 29 to x = 8 29 , we divide both sides by -8: − 8 − 8 x = − 8 − 29 , which simplifies to x = 8 29 . This is not the distributive property.
Conclusion Therefore, the distributive property was applied between Step 1 and Step 2.
Examples
The distributive property is a fundamental concept in algebra and is used in many real-world applications. For example, if you are buying multiple items at a store, each with the same price, you can use the distributive property to calculate the total cost. If you buy 3 items that cost $5 each, the total cost is $3 * $5 = $15. This is a simple application of the distributive property: 3(5) = 15. It's also used in more complex scenarios, such as calculating areas and volumes in geometry or modeling financial investments.
The distributive property was applied between Step 1 and Step 2 when Henry expanded the terms in the equation. Thus, the correct answer is option A: steps 1 and 2. This is where he distributed − 3 and 5 to the terms inside the parentheses.
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