Final answer:
The system of equations 3x + 9y = 27 and x + 3y = 9 is a dependent system with infinitely many solutions, as the two lines are coincident. Any value of y can be used to find a corresponding x using the second equation x = 9 - 3y.
Step-by-step explanation:
To find the solution to the system of equations 3x + 9y = 27 and x + 3y = 9, we can use either the substitution or elimination method. Looking at the second equation, we can notice that x can be expressed in terms of y as x = 9 - 3y. Substituting this into the first equation will give us a single equation with one variable.
Substitute x = 9 - 3y into 3x + 9y = 27:
- 3(9 - 3y) + 9y = 27
- 27 - 9y + 9y = 27
- 27 = 27
We find that the y terms cancel out, leaving us with an identity 27 = 27, which is true for all values of y. This means we have infinitely many solutions, and the system is dependent. To find the corresponding values of x, we can plug any value of y into the second equation x = 9 - 3y to get the corresponding value of x. Thus, the two lines represented by these equations are coincident (one on top of the other).