If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.
6x − 2y = 2,
9x − 3y = 1
There is no solution, since the two equations correspond to parallel lines that do not intersect. In other words, they are incompatible equations.
The second equation can be rewritten
6x - 2y = 2/3, which clearly disagrees with
6x - 2y = 2.
I don't know what you mean by "set builder notation".
To solve the given system of equations:
1. Begin by setting up the system:
6x - 2y = 2 ---- (Equation 1)
9x - 3y = 1 ---- (Equation 2)
2. Notice that the two equations are essentially multiples of each other. Equations that are multiples of each other represent the same line on a graph. Thus, the system of equations has infinitely many solutions.
3. To express the solution set using set-builder notation, we can write it as:
{(x, y) | 6x - 2y = 2}
In this case, the solution set includes all possible (x, y) pairs that satisfy the equation 6x - 2y = 2.
Hence, the solution set using set-builder notation for this system with infinitely many solutions is {(x, y) | 6x - 2y = 2}.