Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
minus, 2, x, plus, 3, y, equals, 3
−2x+3y=
3
minus, 4, x, plus, 7, y, equals, 9
−4x+7y=
9

Answer 1

Compute the determinant of the coefficient matrix:
det = (-2)(7) - (3)(-4) = -14 + 12 = -2 ≠ 0,
so the system has exactly one solution.

Solve by elimination: double the first equation: -4x + 6y = 6. Subtracting from the second gives y = 3. Substitute into -2x + 3y = 3:
-2x + 9 = 3 ⇒ -2x = -6 ⇒ x = 3.

Solution: (x, y) = (3, 3).