Question

drag each tile to the correct location on the table. each tile can be used more than once, but not all tiles will be used.
choose the justification for each step in the solution to the given equation.
subtraction property of equality multiplication property of equality
addition property of equality simplification division property of equality

step	justification
\\(\frac{17}{3} - \frac{3}{4}x - \frac{17}{3} = \frac{1}{2}x + 5 - \frac{17}{3}\\)
\\(-\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3}\\)
\\(-\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x\\)
\\(-\frac{5}{4}x = -\frac{2}{3}\\)
\\(-\frac{5}{4}x \cdot \frac{4}{5} = -\frac{2}{3} \cdot \frac{4}{5}\\)
\\(x = \frac{8}{15}\\)

Explanation:

Step1: Identify the first justification (after given)

In the step \(\frac{17}{3}-\frac{3}{4}x-\frac{17}{3}=\frac{1}{2}x + 5-\frac{17}{3}\), we subtract \(\frac{17}{3}\) from both sides. This is the subtraction property of equality (subtracting the same number from both sides of an equation keeps it equal).

Step2: Identify the second justification

The step \(-\frac{3}{4}x=\frac{1}{2}x-\frac{2}{3}\) is a result of simplifying \(\frac{17}{3}-\frac{17}{3}=0\) and \(5-\frac{17}{3}=\frac{15 - 17}{3}=-\frac{2}{3}\). So this is simplification.

Step3: Identify the third justification

In the step \(-\frac{3}{4}x-\frac{1}{2}x=\frac{1}{2}x-\frac{2}{3}-\frac{1}{2}x\), we subtract \(\frac{1}{2}x\) from both sides. This is the subtraction property of equality.

Step4: Identify the fourth justification

The step \(-\frac{5}{4}x=-\frac{2}{3}\) is a result of simplifying \(-\frac{3}{4}x-\frac{1}{2}x=-\frac{3 + 2}{4}x=-\frac{5}{4}x\) and \(\frac{1}{2}x-\frac{1}{2}x = 0\). So this is simplification.

Step5: Identify the fifth justification

In the step \(-\frac{5}{4}x\cdot\frac{4}{5}=-\frac{2}{3}\cdot\frac{4}{5}\), we multiply both sides by \(\frac{4}{5}\) (the reciprocal of \(-\frac{5}{4}\)). This is the multiplication property of equality.

Step6: Identify the sixth justification

The step \(x=\frac{8}{15}\) is a result of simplifying \(-\frac{5}{4}x\cdot\frac{4}{5}=x\) and \(-\frac{2}{3}\cdot\frac{4}{5}=-\frac{8}{15}\)? Wait, no, \(-\frac{2}{3}\cdot\frac{4}{5}=-\frac{8}{15}\)? Wait, no, \(-\frac{5}{4}x\cdot\frac{4}{5}=x\) and \(-\frac{2}{3}\cdot\frac{4}{5}=-\frac{8}{15}\)? Wait, no, let's recalculate: \(-\frac{2}{3}\times\frac{4}{5}=-\frac{8}{15}\)? Wait, no, the left side: \(-\frac{5}{4}x\times\frac{4}{5}=x\), right side: \(-\frac{2}{3}\times\frac{4}{5}=-\frac{8}{15}\)? Wait, but the final answer is \(x = \frac{8}{15}\). Oh, because both sides are multiplied by \(\frac{4}{5}\), and \(-\frac{2}{3}\times\frac{4}{5}=-\frac{8}{15}\), but then \(x=(-\frac{8}{15})\div(-\frac{5}{4})\)? Wait, no, the step is \(-\frac{5}{4}x\cdot\frac{4}{5}=-\frac{2}{3}\cdot\frac{4}{5}\), so left side: \(-\frac{5}{4}x\times\frac{4}{5}=x\), right side: \(-\frac{2}{3}\times\frac{4}{5}=-\frac{8}{15}\)? But the final answer is \(x=\frac{8}{15}\). Ah, I see, there was a sign error in my previous thought. Wait, the original equation step: \(-\frac{5}{4}x=-\frac{2}{3}\), then multiplying both sides by \(\frac{4}{5}\) (which is the reciprocal of \(-\frac{5}{4}\), but since both sides are negative, the negatives cancel). So \(-\frac{5}{4}x\times\frac{4}{5}=x\), and \(-\frac{2}{3}\times\frac{4}{5}=-\frac{8}{15}\)? No, that can't be. Wait, no, \(-\frac{5}{4}x = -\frac{2}{3}\), multiply both sides by \(\frac{4}{5}\): \(x=(-\frac{2}{3})\times(-\frac{4}{5})=\frac{8}{15}\). Ah, right! Because \(-\frac{5}{4}x\times\frac{4}{5}=x\), and \(-\frac{2}{3}\times\frac{4}{5}\) is wrong, it's \(-\frac{2}{3}\times(-\frac{4}{5})=\frac{8}{15}\). So the justification for \(x = \frac{8}{15}\) is simplification (simplifying the products on both sides).

Now, let's fill the table:

1. For \(\frac{17}{3}-\frac{3}{4}x-\frac{17}{3}=\frac{1}{2}x + 5-\frac{17}{3}\): subtraction property of equality
2. For \(-\frac{3}{4}x=\frac{1}{2}x-\frac{2}{3}\): simplification
3. For \(-\frac{3}{4}x-\frac{1}{2}x=\frac{1}{2}x-\frac{2}{3}-\frac{1}{2}x\): subtraction property of equality
4. For \(-\frac{5}{4}x=-\frac{2}{3}\): simplification
5. For \(-\frac{5}{4}x\cdot\frac{4}{5}=-\frac{2}{3}\cdot\frac{4}{5}\): multiplication property of equality
6. For \(x=\frac{8}{15}\): simplification

Answer:

Step	Justification
\(\frac{17}{3}-\frac{3}{4}x-\frac{17}{3}=\frac{1}{2}x + 5-\frac{17}{3}\)	subtraction property of equality
\(-\frac{3}{4}x=\frac{1}{2}x-\frac{2}{3}\)	simplification
\(-\frac{3}{4}x-\frac{1}{2}x=\frac{1}{2}x-\frac{2}{3}-\frac{1}{2}x\)	subtraction property of equality
\(-\frac{5}{4}x=-\frac{2}{3}\)	simplification
\(-\frac{5}{4}x\cdot\frac{4}{5}=-\frac{2}{3}\cdot\frac{4}{5}\)	multiplication property of equality
\(x=\frac{8}{15}\)	simplification