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Which describes the system of equations below?\newliney=103x−34y = \frac{10}{3}x - \frac{3}{4}\newliney=−27x−83y = -\frac{2}{7}x - \frac{8}{3}\newlineChoices:\newline(A)consistent and dependent\newline(B)consistent and independent\newline(C)inconsistent\newline

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Q. Which describes the system of equations below?\newliney=103x−34y = \frac{10}{3}x - \frac{3}{4}\newliney=−27x−83y = -\frac{2}{7}x - \frac{8}{3}\newlineChoices:\newline(A)consistent and dependent\newline(B)consistent and independent\newline(C)inconsistent\newline
  1. Analyze System of Equations: Analyze the given system of equations to determine if they are consistent, inconsistent, or dependent.\newlineThe system of equations is:\newliney=103x−34y = \frac{10}{3}x - \frac{3}{4}\newliney=−27x−83y = -\frac{2}{7}x - \frac{8}{3}\newlineTo determine the type of system, we need to look at the slopes and yy-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 the first equation, y=103x−34y = \frac{10}{3}x - \frac{3}{4}, the slope (m1m_1) is 103\frac{10}{3} and the yy-intercept (b1b_1) is −34-\frac{3}{4}.\newlineFor the second equation, y=−27x−83y = -\frac{2}{7}x - \frac{8}{3}, the slope (m2m_2) is −27-\frac{2}{7} and the yy-intercept (y=103x−34y = \frac{10}{3}x - \frac{3}{4}11) is y=103x−34y = \frac{10}{3}x - \frac{3}{4}22.\newlineSince the slopes m1m_1 and m2m_2 are different (y=103x−34y = \frac{10}{3}x - \frac{3}{4}55), the lines are not parallel and therefore cannot be dependent.
  3. Determine Consistency: Determine if the system is consistent or inconsistent.\newlineSince the slopes are different, the lines will intersect at exactly one point. This means the system has one solution and is therefore consistent and independent.
  4. Choose Correct Answer: Choose the correct answer based on the analysis.\newlineThe system of equations is consistent and independent because the lines intersect at exactly one point.

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