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How many solutions does the system of equations below have?\newliney=−7x+10y = -7x + 10\newliney=−74x−74y = -\frac{7}{4}x - \frac{7}{4}\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=−7x+10y = -7x + 10\newliney=−74x−74y = -\frac{7}{4}x - \frac{7}{4}\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Analyze Equations: Analyze the given system of equations to determine the number of solutions.\newlineWe have two equations:\newliney=−7x+10y = -7x + 10 (Equation 11)\newliney=−74x−74y = -\frac{7}{4}x - \frac{7}{4} (Equation 22)\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 Parameters: Identify the slopes and y-intercepts of the two lines.\newlineFor Equation 11, the slope m1m_1 is −7-7 and the y-intercept b1b_1 is 1010.\newlineFor Equation 22, the slope m2m_2 is −74-\frac{7}{4} and the y-intercept b2b_2 is −74-\frac{7}{4}.
  3. Compare Slopes: Compare the slopes of the two lines.\newlineIf the slopes are equal and the yy-intercepts are different, the lines are parallel and there is no solution.\newlineIf the slopes are equal and the yy-intercepts are also equal, the lines coincide and there are infinitely many solutions.\newlineIf the slopes are different, the lines intersect at one point and there is one solution.
  4. Determine Solutions: Determine the number of solutions based on the comparison.\newlineSince m1=−7m_1 = -7 and m2=−74m_2 = -\frac{7}{4}, the slopes are not equal. Therefore, the lines are not parallel and will intersect at one point.
  5. Conclude Solution: Conclude the number of solutions for the system of equations.\newlineBecause the lines intersect at one point, there is exactly 11 solution to the system of equations.

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