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How many solutions does the system of equations below have?\newliney=−6x−9y = -6x - 9\newliney=−6x+78y = -6x + \frac{7}{8}\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=−6x−9y = -6x - 9\newliney=−6x+78y = -6x + \frac{7}{8}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Analyze System of Equations: Analyze the given system of equations to determine the number of solutions.\newlineThe system of equations is:\newliney=−6x−9y = -6x - 9\newliney=−6x+78y = -6x + \frac{7}{8}\newlineBoth equations are in the slope-intercept form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. To determine the number of solutions, we need to compare the slopes (mm) and y-intercepts (bb) of the two lines.
  2. Compare Slopes: Compare the slopes of the two equations.\newlineThe slope of the first equation is −6-6, and the slope of the second equation is also −6-6. Since the slopes are equal, the lines are either parallel or the same line.
  3. Compare Y-Intercepts: Compare the y-intercepts of the two equations.\newlineThe y-intercept of the first equation is −9-9, and the y-intercept of the second equation is 78\frac{7}{8}. Since the y-intercepts are different, the lines are parallel and do not intersect.
  4. Conclude Number of Solutions: Conclude the number of solutions based on the comparison of slopes and yy-intercepts.\newlineSince the lines are parallel and have different yy-intercepts, they will never intersect. Therefore, the system of equations has 00 solution.

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