Sure, let's go through the steps one by one, justifying each step 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]
In this step, we used the Distributive Property. This property states that [tex]\(a(b + c) = ab + ac\)[/tex]. So we distribute the multiplication over addition and subtraction inside the parentheses.
3. Combine Like Terms
[tex]\[ 3y + 14 = 2y \][/tex]
Here, we combined like terms. The terms involving [tex]\(y\)[/tex] are [tex]\(6y\)[/tex] and [tex]\(-3y\)[/tex], which combine to [tex]\(3y\)[/tex]. The constant terms [tex]\(8\)[/tex] and [tex]\(6\)[/tex] combine to give [tex]\(14\)[/tex].
4. Isolate Variable Terms on One Side
[tex]\[ 14 = 2y - 3y \][/tex]
We employed the Subtraction Property of Equality, which allows us to subtract the same amount from both sides of an equation. Here, it helps us move the variable term [tex]\(3y\)[/tex] to the other side by subtracting [tex]\(3y\)[/tex] from both sides. This simplifies the equation.
5. Combine Like Terms Again
[tex]\[ 14 = -y \][/tex]
Here, we combine the like terms [tex]\(2y - 3y\)[/tex] to get [tex]\(-y\)[/tex]. This step is another application of combining like terms.
6. Isolate the Variable
[tex]\[ -14 = y \][/tex]
Finally, we use the Multiplication Property of Equality, specifically multiplying both sides by -1, to solve for [tex]\(y\)[/tex].
7. Solution
[tex]\[ y = -14 \][/tex]
Thus, we isolate [tex]\(y\)[/tex] and find the solution to be [tex]\(y = -14\)[/tex].
Putting it all together in the correct order:
1. Original Equation: [tex]\(2(3y + 4) - 3(y - 2) = 2y\)[/tex]
2. Distribute: [tex]\(6y + 8 - 3y + 6 = 2y\)[/tex]
3. Combine Like Terms: [tex]\(3y + 14 = 2y\)[/tex]
4. Subtraction Property of Equality: [tex]\(14 = 2y - 3y\)[/tex]
5. Combine Like Terms: [tex]\(14 = -y\)[/tex]
6. Multiplication Property of Equality: [tex]\(-14 = y\)[/tex]
7. Solution: [tex]\(y = -14\)[/tex]
Each step uses appropriate properties to transform the equation into a simpler form until we isolate [tex]\(y\)[/tex].
