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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. 4, x, plus,...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
Answered by
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**
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