Question
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
3, x, plus, y, equals, minus, 7
3x+y=
−7
minus, 6, x, minus, 2, y, equals, 20
−6x−2y=
20
Answer
Multiple Choice Answers
Infinitely Many SolutionsInfinitely Many Solutions
No SolutionsNo Solutions
One SolutionOne Solution
3, x, plus, y, equals, minus, 7
3x+y=
−7
minus, 6, x, minus, 2, y, equals, 20
−6x−2y=
20
Answer
Multiple Choice Answers
Infinitely Many SolutionsInfinitely Many Solutions
No SolutionsNo Solutions
One SolutionOne Solution
Answers
GPT-4o mini
To determine whether the given system of equations has no solutions, infinitely many solutions, or exactly one solution, we need to analyze the equations. The equations provided are:
1. \(3x + y = -7\)
2. \(-6x - 2y = 20\)
Let's start by rewriting the second equation in a more manageable form:
\[
-6x - 2y = 20
\]
To make it similar in form to the first equation, we can divide the entire equation by \(-2\):
\[
3x + y = -10
\]
Now we have the following two equations:
1. \(3x + y = -7\)
2. \(3x + y = -10\)
Next, we will observe the two equations:
- The first equation states that the sum of \(3x\) and \(y\) equals \(-7\).
- The second equation states that the sum of \(3x\) and \(y\) equals \(-10\).
Since both equations have the same left-hand side (\(3x + y\)), but different right-hand sides (\(-7\) and \(-10\)), these equations represent two parallel lines in a two-dimensional plane. Since parallel lines do not intersect, the system of equations does not have any solutions.
Thus, the conclusion is:
**No Solutions**
1. \(3x + y = -7\)
2. \(-6x - 2y = 20\)
Let's start by rewriting the second equation in a more manageable form:
\[
-6x - 2y = 20
\]
To make it similar in form to the first equation, we can divide the entire equation by \(-2\):
\[
3x + y = -10
\]
Now we have the following two equations:
1. \(3x + y = -7\)
2. \(3x + y = -10\)
Next, we will observe the two equations:
- The first equation states that the sum of \(3x\) and \(y\) equals \(-7\).
- The second equation states that the sum of \(3x\) and \(y\) equals \(-10\).
Since both equations have the same left-hand side (\(3x + y\)), but different right-hand sides (\(-7\) and \(-10\)), these equations represent two parallel lines in a two-dimensional plane. Since parallel lines do not intersect, the system of equations does not have any solutions.
Thus, the conclusion is:
**No Solutions**