Let's justify each step in the process with the appropriate property:
1. Original Equation:
[tex]\[ 2(3y + 4) - 3(y - 2) = 2y \][/tex]
2. Distribute:
[tex]\[ 6y + 8 - 3y + 6 = 2y \][/tex]
Property: Distributive Property of Multiplication over Addition.
(Using the Distributive Property: [tex]\(2 \cdot 3y + 2 \cdot 4\)[/tex] and [tex]\(-3 \cdot y + (-3) \cdot (-2)\)[/tex])
3. Combine Like Terms:
[tex]\[ 3y + 14 = 2y \][/tex]
Property: Combination of Like Terms.
(Combining [tex]\(6y\)[/tex] and [tex]\(-3y\)[/tex] to get [tex]\(3y\)[/tex] and combining [tex]\(8\)[/tex] and [tex]\(6\)[/tex] to get [tex]\(14\)[/tex]).
4. Isolate the Variable Term:
[tex]\[ 14 = -y \][/tex]
Property: Subtract [tex]\(2y\)[/tex] from both sides of the equation.
(Subtracting [tex]\(2y\)[/tex] from [tex]\(3y\)[/tex] results in [tex]\(1y - \)[/tex] or just [tex]\(y\)[/tex]; thus [tex]\(14 = y - 2y\)[/tex]; simplifying [tex]\(14 = -y\)[/tex]).
5. Solve for [tex]\(y\)[/tex]:
[tex]\[ -14 = y \][/tex]
Property: Division Property of Equality (multiplying both sides by [tex]\(-1\)[/tex]).
(Multiplying both sides of the equation [tex]\(14 = -y\)[/tex] by [tex]\(-1\)[/tex] to isolate [tex]\(y\)[/tex] results in [tex]\( -14 = y \)[/tex]).
After filling in the drop-down menus with the correct properties, it looks like this:
\begin{tabular}{rl|l|}
[tex]$2(3 y+4)-3(y-2)$[/tex] & [tex]$=2 y$[/tex] & original equation \\
[tex]$6 y+8-3 y+6$[/tex] & [tex]$=2 y$[/tex] & Distributive Property \\
[tex]$3 y+14$[/tex] & [tex]$=2 y$[/tex] & combining like terms \\
14 & [tex]$=-y$[/tex] & Subtracting [tex]$2y$[/tex] \\
-14 & [tex]$=y$[/tex] & Multiplying by -1
\end{tabular}
This completes the equation solution process with justified steps.
