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How many solutions does the system of equations below have?\newliney=x−5y = x - 5\newliney=−75x+103y = -\frac{7}{5}x + \frac{10}{3}\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=x−5y = x - 5\newliney=−75x+103y = -\frac{7}{5}x + \frac{10}{3}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Analyze Equations: Analyze the given system of equations.\newlineWe have two linear equations:\newline11) y=x−5y = x - 5\newline22) y=−75x+103y = -\frac{7}{5}x + \frac{10}{3}\newlineTo find the number of solutions, we need to determine if the lines are parallel, the same line, or if they intersect at a single point.
  2. Compare Slopes: Compare the slopes of the two lines.\newlineThe slope of the first equation is the coefficient of xx, which is 11.\newlineThe slope of the second equation is the coefficient of xx, which is −75-\frac{7}{5}.\newlineSince the slopes are different (1≠−751 \neq -\frac{7}{5}), the lines are not parallel and must intersect at exactly one point.
  3. Conclude Number of Solutions: Conclude the number of solutions.\newlineBecause the lines are not parallel and have different slopes, they will intersect at exactly one point. Therefore, the system of equations has 11 solution.

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