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How many solutions does the system of equations below have?\newliney=9x+5y = 9x + 5\newliney=9x−34y = 9x - \frac{3}{4}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions

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Determine the number of solutions to a system of equations in three variablesright chevron icon

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Q. How many solutions does the system of equations below have?\newliney=9x+5y = 9x + 5\newliney=9x−34y = 9x - \frac{3}{4}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Analyze System: Analyze the given system of equations to determine the number of solutions.\newlineThe system of equations is:\newliney=9x+5y = 9x + 5\newliney=9x−34y = 9x - \frac{3}{4}\newlineTo find the number of solutions, we need to compare the slopes and yy-intercepts of the two lines represented by these equations.
  2. Identify Slopes and Y-Intercepts: Identify the slopes and y-intercepts of both equations.\newlineFor the first equation, y=9x+5y = 9x + 5, the slope (m1)(m_1) is 99 and the y-intercept (b1)(b_1) is 55.\newlineFor the second equation, y=9x−34y = 9x - \frac{3}{4}, the slope (m2)(m_2) is also 99 and the y-intercept (b2)(b_2) is −34-\frac{3}{4}.
  3. Compare Lines: Compare the slopes and y-intercepts of the two lines.\newlineSince both lines have the same slope m1=m2=9m_1 = m_2 = 9, they are parallel to each other.\newlineHowever, their y-intercepts are different b1=5b_1 = 5 and b2=−34b_2 = -\frac{3}{4}.\newlineParallel lines with different y-intercepts do not intersect and therefore have no points in common.
  4. Conclude Number of Solutions: Conclude the number of solutions based on the comparison.\newlineBecause the lines are parallel and have different yy-intercepts, they will never intersect. Therefore, the system of equations has nono solution.

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