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

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

GPT-4o mini · Answer

To determine whether the system of equations has no solutions, infinitely many solutions, or exactly one solution, we can analyze the two equations:

$- 5 x + 4 y = - 7 (1)$ $17 x - 16 y = 31 (2)$

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):

$4 y = 5 x - 7$

Dividing by 4: $y = \frac{5}{4} x - \frac{7}{4}$

Transforming Equation (2):

$- 16 y = - 17 x + 31$

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