The system x - 6y = 4, 3x - 18y = 4 has no solution
Change one constant or coefficient to make a new system with an infinite number of solutions.
To make the system have an infinite number of solutions, we need to create a situation where the two equations are dependent (meaning one equation can be obtained by multiplying the other equation by a constant). To do this, we can change the second equation slightly.
Let's modify the second equation by multiplying it by 2:
Original system:
1) x - 6y = 4
2) 3x - 18y = 4
Modified system:
1) x - 6y = 4
2) 6x - 36y = 8
Now, if we compare the two equations, we can see that the second equation is simply the first equation multiplied by 6. Therefore, the two equations are dependent and there will be an infinite number of solutions.
To change one constant or coefficient in the system x - 6y = 4, 3x - 18y = 4 in order to obtain a new system with an infinite number of solutions, we need to create a dependent equation. This can be achieved by making the two equations proportional to each other.
Let's multiply the first equation by 3. This will result in the new system:
3x - 18y = 12, 3x - 18y = 4.
Now, we have two equations that are identical. This means they represent the same line in the coordinate plane, and therefore, there are an infinite number of solutions to this system.
Your system has no solution because you have two parallel lines
To have an infinite number of solutions, the two equations must be one and the same
notice the left side of your second equation is 3 times the left side of your first equation,
so what do you think should happen to the right side ??