Question

drag each step and justification to the correct location on the table. each step and justification can be used more than once, but not all steps and justifications will be used. order each step and justification that is needed to solve the equation below. \\(\frac{2}{3}y + 15 = 9\\) \\(\frac{2}{3}y \cdot \frac{3}{2} = -6 \cdot \frac{3}{2}\\) \\(\frac{2}{3}y \cdot \frac{3}{2} = 6 \cdot \frac{3}{2}\\) \\(y = 9\\) \\(\frac{2}{3}y = -6\\) multiplication property of equality \\(y = -9\\) subtraction property of equality \\(\frac{2}{3}y = 6\\) \\(\

$$\begin{array}{|c|c|} \\hline \\textbf{steps} & \\textbf{justifications} \\\\ \\hline \\frac{2}{3}y + 15 = 9 & \\text{given} \\\\ \\hline \\frac{2}{3}y + 15 - 15 = 9 - 15 & \\\\ \\hline & \\text{simplification} \\\\ \\hline & \\\\ \\hline & \\text{simplification} \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Apply Subtraction Property of Equality

To isolate the term with \( y \), we subtract 15 from both sides of the equation \( \frac{2}{3}y + 15 = 9 \). By the subtraction property of equality, if \( a = b \), then \( a - c = b - c \). So we have \( \frac{2}{3}y + 15 - 15 = 9 - 15 \). The justification for this step is the "Subtraction property of equality".

Step2: Simplify the Equation

Simplify both sides of \( \frac{2}{3}y + 15 - 15 = 9 - 15 \). The \( +15 - 15 \) on the left cancels out, and \( 9 - 15 = -6 \), so we get \( \frac{2}{3}y = -6 \). This step is a simplification.

Step3: Apply Multiplication Property of Equality

To solve for \( y \), we multiply both sides of \( \frac{2}{3}y = -6 \) by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). By the multiplication property of equality, if \( a = b \), then \( a \cdot c = b \cdot c \) (where \( c
eq0 \)). So we have \( \frac{2}{3}y \cdot \frac{3}{2} = -6 \cdot \frac{3}{2} \). The justification for this step is the "Multiplication property of equality".

Step4: Simplify to Find \( y \)

Simplify both sides of \( \frac{2}{3}y \cdot \frac{3}{2} = -6 \cdot \frac{3}{2} \). The \( \frac{2}{3} \) and \( \frac{3}{2} \) on the left cancel out, and \( -6 \cdot \frac{3}{2}=-9 \), so we get \( y = -9 \). This step is a simplification.

Now, filling in the table:

Steps Column:

1. \( \frac{2}{3}y + 15 = 9 \) (Given)
2. \( \frac{2}{3}y + 15 - 15 = 9 - 15 \)
3. \( \frac{2}{3}y = -6 \)
4. \( \frac{2}{3}y \cdot \frac{3}{2} = -6 \cdot \frac{3}{2} \)
5. \( y = -9 \)

Justifications Column:

1. Given
2. Subtraction property of equality
3. Simplification
4. Multiplication property of equality
5. Simplification

So the completed table (with steps and justifications in order) is:

Steps	Justifications
\( \frac{2}{3}y + 15 - 15 = 9 - 15 \)	Subtraction property of equality
\( \frac{2}{3}y = -6 \)	Simplification
\( \frac{2}{3}y \cdot \frac{3}{2} = -6 \cdot \frac{3}{2} \)	Multiplication property of equality
\( y = -9 \)	Simplification

Answer:

The completed table with correct steps and justifications is as shown above. The key steps are using the subtraction property of equality to isolate the term with \( y \), simplifying, using the multiplication property of equality, and then simplifying again to find \( y = -9 \).