Question

step statements reasons 1 $-6 = -\frac{2}{3}(x + 12) + \frac{1}{3}x$ given 2 $-6 = -\frac{2}{3}x - 8 + \frac{1}{3}x$ 3 $-6 = -\frac{1}{3}x - 8$ 4 $2 = -\frac{1}{3}x$ 5 $-6 = x$ 6 $x = -6$ distributive property addition property of equality subtraction property of equality multiplication property of equality transitive property combine like terms symmetric property

Explanation:

Step 1: Identify Reason for Step 2

To get from \(-6 = -\frac{2}{3}(x + 12) + \frac{1}{3}x\) to \(-6 = -\frac{2}{3}x - 8 + \frac{1}{3}x\), we use the distributive property. The distributive property states that \(a(b + c) = ab + ac\). Here, \(a = -\frac{2}{3}\), \(b = x\), and \(c = 12\), so \(-\frac{2}{3}(x + 12) = -\frac{2}{3}x - \frac{2}{3} \times 12 = -\frac{2}{3}x - 8\).

Step 2: Identify Reason for Step 3

To get from \(-6 = -\frac{2}{3}x - 8 + \frac{1}{3}x\) to \(-6 = -\frac{1}{3}x - 8\), we combine like terms. The terms \(-\frac{2}{3}x\) and \(\frac{1}{3}x\) are like terms (both have \(x\) with coefficient involving \(\frac{1}{3}\)). Combining them: \(-\frac{2}{3}x + \frac{1}{3}x = -\frac{1}{3}x\).

Step 3: Identify Reason for Step 4

To get from \(-6 = -\frac{1}{3}x - 8\) to \(2 = -\frac{1}{3}x\), we use the addition property of equality. The addition property of equality states that if \(a = b\), then \(a + c = b + c\). Here, we add 8 to both sides: \(-6 + 8 = -\frac{1}{3}x - 8 + 8\), which simplifies to \(2 = -\frac{1}{3}x\).

Step 4: Identify Reason for Step 5

To get from \(2 = -\frac{1}{3}x\) to \(-6 = x\), we use the multiplication property of equality. The multiplication property of equality states that if \(a = b\), then \(a \times c = b \times c\) (for \(c
eq 0\)). Here, we multiply both sides by \(-3\): \(2 \times (-3) = -\frac{1}{3}x \times (-3)\), which simplifies to \(-6 = x\).

Step 5: Identify Reason for Step 6

To get from \(-6 = x\) to \(x = -6\), we use the symmetric property. The symmetric property states that if \(a = b\), then \(b = a\). So if \(-6 = x\), then \(x = -6\).

Answer:

• Step 2 Reason: distributive property
• Step 3 Reason: combine like terms
• Step 4 Reason: addition property of equality
• Step 5 Reason: multiplication property of equality
• Step 6 Reason: symmetric property