Final answer:
The equivalent systems of equations to the original system given are options (a) and (c). These systems are scaled versions of the original, which does not change the solution set of the equations.
Step-by-step explanation:
To determine which system of equations is equivalent to 5x + 6y = 12 and 7x - 9y = 11, we can look for transformations that keep the equations' solutions unchanged. An equivalent system will be one where the equations represent the same lines in a Cartesian plane, thereby having the same solutions for x and y.
Looking at option (a) 10x + 12y = 24 and 7x - 9y = 11, we see the first equation is simply the original first equation multiplied by 2. This doesn't change the solution set of the equation, hence the two systems are equivalent.
Option (b) is not equivalent as the second equation 21x - 9y = 33 cannot be derived by simply scaling the second original equation by a constant.
Option (c) has the first equation multiplied by -3, -15x - 18y = 36, which again has no impact on the solution set and thus is equivalent. However, as the second equation remains unchanged, we can consider it as the original system.
Lastly, option (d) has 5x + 6y = 12 and 14x - 18y = -22. The second equation is not a scalar multiple of the original second equation, therefore it's not equivalent.
In conclusion, options (a) and (c) are equivalent systems to the original one, as they represent the same set of solutions. Hence, option (a) 10x + 12y = 24 and 7x - 9y = 11, and option (c) -15x - 18y = 36 and 7x - 9y = 11 are equivalent to the initial system of equations.