Final answer:
The given equations are equivalent to the original equation 3(x + 4) = 7 because they are derived from it by applying the properties of equality, such as distribution, multiplication, addition, and the symmetric property, which do not change the solution set of the equation.
Step-by-step explanation:
All the given equations are equivalent to the original equation 3(x + 4) = 7 and thus have the same solutions. This is because they can be derived from the original equation through valid algebraic operations, a concept known as the properties of equality. These properties allow for certain manipulations of an equation that change its appearance without changing its solution set.
- 3x + 12 = 7 is obtained by distributing the 3 across the parentheses in the original equation.
- 6(x + 4) = 14 is the result of multiplying both sides of the original equation by 2, which is a valid operation as long as it's applied to both sides of the equation (Multiplication Property of Equality).
- 3(x + 4) - 5 = 2 can be turned back into the original equation by adding 5 to both sides, thus utilizing the Addition Property of Equality.
- 7 = 3(x + 4) is simply the original equation with both sides switched. The Symmetric Property of Equality states that if a = b, then b = a, which does not change the solutions.
Therefore, each of these equations will yield the same solution when solved.