Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
2, x, plus, y, equals, 6
2x+y=
6
minus, 4, x, minus, y, equals, minus, 4
−4x−y=−4

Answer 1

To determine the number of solutions for the system of equations, we can examine the coefficients of x and y.

The first equation is:
2x + y = 6

The second equation is:
-4x - y = -4

To solve this system of equations, we can use the method of elimination. We need to eliminate either x or y to find the value of the remaining variable.

To eliminate y:
Multiply the first equation by -1:
-2x - y = -6

Add the two equations:
(-2x - y) + (-4x - y) = -6 + (-4)
-6x - 2y = -10

Now we have the equation:
-6x - 2y = -10

Since the coefficients of both x and y are non-zero, the system of equations does not have infinitely many solutions.

Since the coefficients of x and y are not equal in this new equation, and the equation is not an identity (0 = 0), the system of equations does not have exactly one solution.

Therefore, the system of equations has NO SOLUTIONS.