Final answer:
Each of the given equations (a, b, c, d) represents the same solutions as 3(x + 4) = 7 due to the properties of equality and equivalent transformations, including the Distributive, Multiplication and Subtraction Properties of Equality, and the Symmetric Property of Equality.
Step-by-step explanation:
The equations provided each have the same solutions as the original equation 3(x + 4) = 7 because of the properties of equality and equivalent transformations of equations. Let's explain for each alternative:
- a. 3x + 12 = 7 is obtained by applying the Distributive Property on the original equation, which is one of the reasons why they have the same solution.
- b. 6(x + 4) = 14 is the original equation multiplied by 2 on both sides. This is an example of the Multiplication Property of Equality.
- c. 3(x+4) - 5 = 2 is the original equation with 5 subtracted from both sides. This demonstrates the Subtraction Property of Equality.
- d. 7 = 3(x + 4) is just the original equation with both sides switched. This does not change the solution because of the Symmetric Property of Equality.
To solve these equations, one can simplify and solve for x using algebraic steps such as combining like terms and isolating the variable. Checking for reasonableness is advised after finding the solution.