Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution. 2x − y = 2 3x + y = −6 one and only one solution infinitely many solutions no solution find the solution, if one exists. (if there are infinitely many solutions, express x and y in terms of the parameter t. if there is no solution, enter no solution.) (x, y) =

Step-by-step explanation:

You know there is only one solution because the ratio of x- and y-coefficients is different in the two equations. That means the lines will have different slopes, so must intersect in exactly one point.

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The y-coefficients are opposites, so you can eliminate the y-variable by adding the equations:

  (2x -y) + (3x +y) = (2) + (-6)

  5x = -4

  x = -4/5 = -0.8

Substituting this into the second equation, we have ...

  3(-0.8) +y = -6

  y = -3.6 . . . . . . . add 2.4 to both sides

The solution is (x, y) = (-0.8, -3.6).

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You can also find the solution by graphing (or using a graphing calculator).

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