Question
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
minus, 2, x, plus, y, equals, minus, 3
−2x+y=
−3
6, x, minus, 3, y, equals, 7
6x−3y=
7
minus, 2, x, plus, y, equals, minus, 3
−2x+y=
−3
6, x, minus, 3, y, equals, 7
6x−3y=
7
Answers
GPT 3.5
To determine the number of solutions, we can rewrite the system of equations in slope-intercept form:
-2x + y = -3 (Equation 1)
6x - 3y = 7 (Equation 2)
For Equation 1:
y = 2x - 3
For Equation 2:
2x - y = -7
-3y = -7 - 6x
y = 2x + (7/3)
Comparing the slopes of these equations, we see that they are equal (both are 2). However, the y-intercepts (-3 for Equation 1 and 7/3 for Equation 2) are not equal.
Since the slopes are equal and the y-intercepts are not equal, the system of equations represents two parallel lines. Therefore, there are no solutions, so the system has no solutions.
-2x + y = -3 (Equation 1)
6x - 3y = 7 (Equation 2)
For Equation 1:
y = 2x - 3
For Equation 2:
2x - y = -7
-3y = -7 - 6x
y = 2x + (7/3)
Comparing the slopes of these equations, we see that they are equal (both are 2). However, the y-intercepts (-3 for Equation 1 and 7/3 for Equation 2) are not equal.
Since the slopes are equal and the y-intercepts are not equal, the system of equations represents two parallel lines. Therefore, there are no solutions, so the system has no solutions.