Question

shena made a table to show the steps and algebraic properties used to write an equivalent expression for 3x−5+7x+6.

step	expression	property
2	(3x + 7x) + (6 − 5)	distributive property
3	(3 + 7)x + (6 − 5)	distributive property
4	10x + 1	addition property

what error did shena make?

• shena should have written the associative property for step 1.
• shena should have written the associative property for step 2.
• shena should have written the associative property for step 3.
• shena should have written the associative property for step 4.

Explanation:

Step1: Analyze Step 3

The expression in Step 3 is \((3 + 7)x+(6 - 5)\). The operation here is combining like terms (combining the \(x\)-terms and the constant terms), which is using the distributive property in reverse (factoring out the common factor for the \(x\)-terms) and then combining constants. But actually, for combining \(3x+7x\) and \(6 - 5\), the associative property of addition (grouping terms for addition) should be considered, not the distributive property. Wait, no, let's re - examine. Wait, Step 3: The original expression after Step 2 is \((3x + 7x)+(6 - 5)\). To get to \((3 + 7)x+(6 - 5)\), we use the distributive property \(ax+bx=(a + b)x\) for the \(x\)-terms. But the property labeled is distributive property, but let's check the options. The options are about when she should have used the associative property. Wait, Step 3: The expression transformation from Step 2 \((3x + 7x)+(6 - 5)\) to Step 3 \((3 + 7)x+(6 - 5)\) is actually the distributive property (factoring \(x\) from \(3x+7x\)). But the error is in the property label? No, the options are about when she should have written the associative property. Wait, let's check each step:

• Step 1: \(3x+7x + 6-5\) (from \(3x-5 + 7x+6\) using commutative property, which is correct as we can reorder terms: \(3x+7x+6 - 5\) from \(3x-5+7x + 6\) by commuting \(-5\) and \(7x+6\))
• Step 2: \((3x + 7x)+(6 - 5)\) (using associative property to group \(3x+7x\) and \(6 - 5\) for addition, but the property labeled is distributive, which is wrong. Wait, no, the property for Step 2 is labeled as distributive, but it should be associative. Wait, the options are:

The options are:

1. Shena should have written the associative property for step 1.

1. Shena should have written the associative property for step 2.

1. Shena should have written the associative property for step 3.

1. Shena should have written the associative property for step 4.

Wait, Step 2: The transformation from Step 1 \(3x+7x + 6-5\) to Step 2 \((3x + 7x)+(6 - 5)\) is using the associative property of addition (grouping the \(x\)-terms together and the constant terms together for addition). But the property labeled for Step 2 is distributive property. Wait, no, the property column for Step 2 is "Distributive property", but it should be associative property. Wait, the question is "What error did Shena make?" and the options are about when she should have used the associative property. So Step 2: The operation of grouping \((3x + 7x)\) and \((6 - 5)\) is the associative property of addition (since we are grouping terms to add: \((a + b)+(c + d)=a + b + c + d\) and here we group \((3x+7x)\) and \((6 - 5)\)). But Shena labeled Step 2 with distributive property, but she should have used associative property for Step 2. Wait, the options are "Shena should have written the associative property for step 2." So that's the error.

Step 2: Conclusion

So the error is that in Step 2, she used the distributive property label, but she should have used the associative property. So the correct option is "Shena should have written the associative property for step 2."

Answer:

Shena should have written the associative property for step 2. (The option corresponding to "Shena should have written the associative property for step 2.")