Question
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
x, plus, y, equals, minus, 8
x+y=
−8
2, x, plus, 2, y, equals, minus, 16
2x+2y=
−16
x, plus, y, equals, minus, 8
x+y=
−8
2, x, plus, 2, y, equals, minus, 16
2x+2y=
−16
Answers
GPT 3.5
To determine the number of solutions for the system of equations, we can use two methods: graphing or algebraic manipulation.
1) Graphing Method:
Graphing the two equations, we can see that they have the same slope and different y-intercepts. This means that the lines are parallel and will never intersect. Therefore, there are no solutions for this system of equations.
2) Algebraic Manipulation Method:
We'll rewrite the equations in standard form:
1) x + y = -8 --> x + y + 8 = 0
2) 2x + 2y = -16 --> 2x + 2y + 16 = 0
If we divide equation 2 by 2, we get:
x + y + 8 = 0
This is the same equation as equation 1, so the two equations represent the same line. Therefore, they have infinitely many solutions.
1) Graphing Method:
Graphing the two equations, we can see that they have the same slope and different y-intercepts. This means that the lines are parallel and will never intersect. Therefore, there are no solutions for this system of equations.
2) Algebraic Manipulation Method:
We'll rewrite the equations in standard form:
1) x + y = -8 --> x + y + 8 = 0
2) 2x + 2y = -16 --> 2x + 2y + 16 = 0
If we divide equation 2 by 2, we get:
x + y + 8 = 0
This is the same equation as equation 1, so the two equations represent the same line. Therefore, they have infinitely many solutions.