Answer:
- infinitely many
- no solution
- one solution
Explanation:
You want to determine the number of solutions to three different systems of equations.
Standard form
A linear equation is written in standard form when the coefficients are mutually prime, and the leading coefficient is positive:
ax +by = c . . . . . . GCF(a, b, c) = 1, a > 0
The equations are easiest to compare when they are all written in standard form.
Numbers of solutions
A system will have an infinite number of solutions when the equations are identical.
A system will have zero solutions when it reduces to ...
(non-zero constant) = 0
A system will have one solution when the equations are different.
System 1
A factor of 2 can be removed from the first equation:
2x -3y = 5
A factor of 3 can be removed from the second equation:
2x -3y = 5
These equations are identical, so have infinitely many solutions.
System 2
Multiplying the first equation by 2 gives ...
2y = -3x +6
Adding 3x, we have ...
3x +2y = 6
When we subtract the second equation from this, we get ...
(3x +2y) -(3x +2y) = (6) -(3)
0 = 3
These equations have no solution.
System 3
These equations are already in standard form, and are different. This system has one solution.
(The exact solution is (x, y) = (0.48, 3.36).)