Question

question
choose the justification for each step in the solution of the given equation.
drag the labels to the correct locations on the table. not all labels will be used.

step\tstatements\treasons
1\t-6 = -\frac{2}{3}(x + 12) + \frac{1}{3}x\tgiven
2\t-6 = -\frac{2}{3}x - 8 + \frac{1}{3}x\t
3\t-6 = -\frac{1}{3}x - 8\t
4\t2 = -\frac{1}{3}x\t
5\t-6 = x\t
6\tx = -6\t

distributive property\taddition property of equality\tsubtraction property of equality
multiplication property of equality\ttransitive property\tcombine like terms\tsymmetric property

Explanation:

To solve this, we analyze each step:

Step 2:

• Explanation: We apply the distributive property to expand \(-\frac{2}{3}(x + 12)\). The distributive property states \(a(b + c)=ab + ac\), so \(-\frac{2}{3}(x + 12)=-\frac{2}{3}x-\frac{2}{3}(12)=-\frac{2}{3}x - 8\).
• Reason: distributive property

Step 3:

• Explanation: Combine the like terms \(-\frac{2}{3}x\) and \(\frac{1}{3}x\). \(-\frac{2}{3}x+\frac{1}{3}x=-\frac{1}{3}x\), so we simplify the equation.
• Reason: combine like terms

Step 4:

• Explanation: Add 8 to both sides of the equation (addition property of equality: if \(a = b\), then \(a + c = b + c\)). Adding 8 to both sides of \(-6=-\frac{1}{3}x - 8\) gives \(-6 + 8=-\frac{1}{3}x-8 + 8\), so \(2=-\frac{1}{3}x\).
• Reason: addition property of equality

Step 5:

• Explanation: Multiply both sides by \(-3\) (multiplication property of equality: if \(a = b\), then \(ac = bc\)). Multiplying \(2=-\frac{1}{3}x\) by \(-3\) gives \(2(-3)=-\frac{1}{3}x(-3)\), so \(-6 = x\).
• Reason: multiplication property of equality

Step 6:

• Explanation: Use the symmetric property, which states if \(a = b\), then \(b = a\). Since \(-6 = x\), we can rewrite it as \(x = -6\).
• Reason: symmetric property

Final Table (Reasons Filled):

Step	Statements	Reasons
2	\(-6 = -\frac{2}{3}x - 8+\frac{1}{3}x\)	distributive property
3	\(-6 = -\frac{1}{3}x - 8\)	combine like terms
4	\(2 = -\frac{1}{3}x\)	addition property of equality
5	\(-6 = x\)	multiplication property of equality
6	\(x = -6\)	symmetric property

(Note: The labels are dragged to match the reasons above.)

Answer:

To solve this, we analyze each step:

Step 2:

• Explanation: We apply the distributive property to expand \(-\frac{2}{3}(x + 12)\). The distributive property states \(a(b + c)=ab + ac\), so \(-\frac{2}{3}(x + 12)=-\frac{2}{3}x-\frac{2}{3}(12)=-\frac{2}{3}x - 8\).
• Reason: distributive property

Step 3:

• Explanation: Combine the like terms \(-\frac{2}{3}x\) and \(\frac{1}{3}x\). \(-\frac{2}{3}x+\frac{1}{3}x=-\frac{1}{3}x\), so we simplify the equation.
• Reason: combine like terms

Step 4:

• Explanation: Add 8 to both sides of the equation (addition property of equality: if \(a = b\), then \(a + c = b + c\)). Adding 8 to both sides of \(-6=-\frac{1}{3}x - 8\) gives \(-6 + 8=-\frac{1}{3}x-8 + 8\), so \(2=-\frac{1}{3}x\).
• Reason: addition property of equality

Step 5:

• Explanation: Multiply both sides by \(-3\) (multiplication property of equality: if \(a = b\), then \(ac = bc\)). Multiplying \(2=-\frac{1}{3}x\) by \(-3\) gives \(2(-3)=-\frac{1}{3}x(-3)\), so \(-6 = x\).
• Reason: multiplication property of equality

Step 6:

• Explanation: Use the symmetric property, which states if \(a = b\), then \(b = a\). Since \(-6 = x\), we can rewrite it as \(x = -6\).
• Reason: symmetric property

Final Table (Reasons Filled):

Step	Statements	Reasons
2	\(-6 = -\frac{2}{3}x - 8+\frac{1}{3}x\)	distributive property
3	\(-6 = -\frac{1}{3}x - 8\)	combine like terms
4	\(2 = -\frac{1}{3}x\)	addition property of equality
5	\(-6 = x\)	multiplication property of equality
6	\(x = -6\)	symmetric property

(Note: The labels are dragged to match the reasons above.)