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Which describes the system of equations below?\newliney=−3x−4y = -3x - 4\newliney=83x+52y = \frac{8}{3}x + \frac{5}{2}\newlineChoices:\newline(A)inconsistent\newline(B)consistent and independent\newline(C)consistent and dependent

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Q. Which describes the system of equations below?\newliney=−3x−4y = -3x - 4\newliney=83x+52y = \frac{8}{3}x + \frac{5}{2}\newlineChoices:\newline(A)inconsistent\newline(B)consistent and independent\newline(C)consistent and dependent
  1. Analyze System Type: Analyze the given system of equations to determine its type.\newlineWe have two equations:\newliney=−3x−4y = -3x - 4 (Equation 11)\newliney=83x+52y = \frac{8}{3}x + \frac{5}{2} (Equation 22)\newlineTo determine if the system is consistent and independent, consistent and dependent, or inconsistent, we need to compare the slopes and y-intercepts of the two lines represented by these equations.
  2. Identify Slopes and Intercepts: Identify the slopes and yy-intercepts of the two lines.\newlineFor Equation 11, the slope (m1m_1) is −3-3 and the yy-intercept (b1b_1) is −4-4.\newlineFor Equation 22, the slope (m2m_2) is 83\frac{8}{3} and the yy-intercept (b2b_2) is m1m_100.
  3. Compare Slopes: Compare the slopes of the two lines.\newlineIf the slopes are equal and the yy-intercepts are also equal, the system is consistent and dependent (the lines are the same).\newlineIf the slopes are equal and the yy-intercepts are different, the system is inconsistent (the lines are parallel and never intersect).\newlineIf the slopes are different, the system is consistent and independent (the lines intersect at one point).
  4. Determine System Type: Determine the type of system based on the slopes and y-intercepts.\newlineSince m1=−3m_1 = -3 and m2=83m_2 = \frac{8}{3}, the slopes are not equal.\newlineTherefore, the system is consistent and independent because the lines will intersect at exactly one point.

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