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How many solutions does the system of equations below have?\newliney=5x+10y = 5x + 10\newliney=54x+95y = \frac{5}{4}x + \frac{9}{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=5x+10y = 5x + 10\newliney=54x+95y = \frac{5}{4}x + \frac{9}{5}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Compare Slopes and Y-Intercepts: To determine the number of solutions for the system of equations, we need to compare the slopes and y-intercepts of the two lines represented by the equations. If the slopes are different, the lines will intersect at one point, indicating one solution. If the slopes are the same but the y-intercepts are different, the lines are parallel and there is no solution. If both the slopes and y-intercepts are the same, the lines coincide and there are infinitely many solutions.
  2. Identify First Equation: First, let's identify the slope and y-intercept of the first equation, y=5x+10y = 5x + 10. The slope is 55 and the y-intercept is 1010.
  3. Identify Second Equation: Now, let's identify the slope and y-intercept of the second equation, y=54x+95y = \frac{5}{4}x + \frac{9}{5}. The slope is 54\frac{5}{4} and the y-intercept is 95\frac{9}{5}.
  4. Compare Slopes: Comparing the slopes of the two equations, we see that the first equation has a slope of 55, and the second equation has a slope of 54\frac{5}{4}. Since 55 is not equal to 54\frac{5}{4}, the slopes are different.
  5. Determine Number of Solutions: Because the slopes are different, the lines will intersect at exactly one point. Therefore, the system of equations has 11 solution.

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