Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.

Question

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

Answer 1

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