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How many solutions does the system of equations below have?\newliney=−27x−7y = -\frac{2}{7}x - 7\newliney=−27x+75y = -\frac{2}{7}x + \frac{7}{5}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions

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Q. How many solutions does the system of equations below have?\newliney=−27x−7y = -\frac{2}{7}x - 7\newliney=−27x+75y = -\frac{2}{7}x + \frac{7}{5}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Analyze slopes: Analyze the slopes of both equations.\newlineThe slope of the first equation y=−27x−7y = \frac{-2}{7}x - 7 is −27-\frac{2}{7}.\newlineThe slope of the second equation y=−27x+75y = \frac{-2}{7}x + \frac{7}{5} is also −27-\frac{2}{7}.\newlineSince both slopes are equal, the lines are either parallel or the same line.
  2. Compare y-intercepts: Compare the y-intercepts of both equations.\newlineThe y-intercept of the first equation is −7-7.\newlineThe y-intercept of the second equation is 75\frac{7}{5}.\newlineSince the y-intercepts are different, the lines are parallel and do not intersect.
  3. Determine solutions: Determine the number of solutions.\newlineParallel lines never intersect, so there are no points that satisfy both equations simultaneously.\newlineTherefore, the system of equations has no solution.

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