Question
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
minus, 5, x, plus, 4, y, equals, minus, 7
−5x+4y=
−7
17, x, minus, 16, y, equals, 31
17x−16y=
31
Answer
Multiple Choice Answers
Infinitely Many SolutionsInfinitely Many Solutions
One SolutionOne Solution
No SolutionsNo Solutions
minus, 5, x, plus, 4, y, equals, minus, 7
−5x+4y=
−7
17, x, minus, 16, y, equals, 31
17x−16y=
31
Answer
Multiple Choice Answers
Infinitely Many SolutionsInfinitely Many Solutions
One SolutionOne Solution
No SolutionsNo Solutions
Answers
Answered by
GPT-4o mini
To determine whether the system of equations has no solutions, infinitely many solutions, or exactly one solution, we can analyze the two equations:
\[
-5x + 4y = -7 \quad \text{(1)}
\]
\[
17x - 16y = 31 \quad \text{(2)}
\]
We can use the elimination or substitution method. However, a more straightforward approach to analyze the system is to rewrite both equations in slope-intercept form (y = mx + b).
### Transforming Equation (1):
\[
4y = 5x - 7
\]
Dividing by 4:
\[
y = \frac{5}{4}x - \frac{7}{4}
\]
### Transforming Equation (2):
\[
-16y = -17x + 31
\]
Dividing by -16:
\[
y = \frac{17}{16}x - \frac{31}{16}
\]
### Finding the Slope:
From the first equation, the slope \(m_1 = \frac{5}{4}\).
From the second equation, the slope \(m_2 = \frac{17}{16}\).
### Checking Slopes:
Since \(m_1 \neq m_2\), the two lines are not parallel, meaning they will intersect at exactly one point.
### Conclusion:
The system of equations has exactly one solution.
Therefore, the answer is:
**One Solution**
\[
-5x + 4y = -7 \quad \text{(1)}
\]
\[
17x - 16y = 31 \quad \text{(2)}
\]
We can use the elimination or substitution method. However, a more straightforward approach to analyze the system is to rewrite both equations in slope-intercept form (y = mx + b).
### Transforming Equation (1):
\[
4y = 5x - 7
\]
Dividing by 4:
\[
y = \frac{5}{4}x - \frac{7}{4}
\]
### Transforming Equation (2):
\[
-16y = -17x + 31
\]
Dividing by -16:
\[
y = \frac{17}{16}x - \frac{31}{16}
\]
### Finding the Slope:
From the first equation, the slope \(m_1 = \frac{5}{4}\).
From the second equation, the slope \(m_2 = \frac{17}{16}\).
### Checking Slopes:
Since \(m_1 \neq m_2\), the two lines are not parallel, meaning they will intersect at exactly one point.
### Conclusion:
The system of equations has exactly one solution.
Therefore, the answer is:
**One Solution**
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