A linear system may have a unique solution, no solution, or infinitely many solutions. Indicate the type of the system for th following examples by U , N , or I , respectively.

1. 2x+3y=5 4x+6y=10
2. 2x+3y=5 2x+4y=6
3. 2x+3y=5 2x+3y=6

1. One equation is twice the other, but represents the same line. There are an infinite number of (x,y) combinations that satisfy it.

2. Those lines intercept at a point, for one solution.

3. Both equations cannot be true. Those are the equations of parallel lines

1. I (infinitely many solutions) - The two equations are equivalent and represent the same line.

2. N (no solution) - The two equations represent parallel lines that do not intersect.
3. N (no solution) - The two equations represent parallel lines that do not intersect.

To determine the type of the system for each example, we can use a method like Gaussian elimination or matrix operations to analyze the system of equations. However, we can also identify the type of the system by examining the coefficients and constants directly.

1. 2x+3y=5 4x+6y=10
We notice that the second equation is a multiple of the first equation. Since the equations represent two parallel lines (or the same line), the system has infinitely many solutions. Therefore, the type of this system is I.

2. 2x+3y=5 2x+4y=6
We can subtract the equations to eliminate x:
(2x+3y) - (2x+4y) = 5 - 6
-y = -1
y = 1
Now, we can substitute the value of y back into either equation to find x:
2x + 3(1) = 5
2x + 3 = 5
2x = 5 - 3
2x = 2
x = 1
The system has a unique solution (x=1, y=1). Therefore, the type of this system is U.

3. 2x+3y=5 2x+3y=6
Subtracting the equations to eliminate x:
(2x+3y) - (2x+3y) = 5 - 6
0 = -1
This is a contradiction (0 cannot equal -1). Therefore, the system has no solution. The type of this system is N.

In summary:
1. Type of system: I (infinitely many solutions)
2. Type of system: U (unique solution)
3. Type of system: N (no solution)