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Which of the following equations represents a line that passes through the points (0,-1) and (-9,2) ? I. y+2=(1)/(3)(x-3) II. 2x+6y=-6 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (0,−1) (0,-1) and (−9,2) (-9,2) ?\newlineI. y+2=13(x−3) y+2=\frac{1}{3}(x-3) \newlineII. 2x+6y=−6 2 x+6 y=-6 \newlineNeither\newlineI only\newlineII only\newlineI and II

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Q. Which of the following equations represents a line that passes through the points (0,−1) (0,-1) and (−9,2) (-9,2) ?\newlineI. y+2=13(x−3) y+2=\frac{1}{3}(x-3) \newlineII. 2x+6y=−6 2 x+6 y=-6 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate Slope: Find the slope of the line passing through the points (0,−1)(0,-1) and (−9,2)(-9,2). The slope mm is calculated using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. m=2−(−1)−9−0m = \frac{2 - (-1)}{-9 - 0} m=2+1−9m = \frac{2 + 1}{-9} m=3−9m = \frac{3}{-9} m=−13m = -\frac{1}{3}
  2. Write Point-Slope Equation: Use the slope and one of the points to write the equation of the line in point-slope form.\newlineWe can use the point (0,−1)(0,-1) and the slope −13-\frac{1}{3}.\newlineThe point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1).\newliney−(−1)=(−13)(x−0)y - (-1) = (-\frac{1}{3})(x - 0)\newliney+1=(−13)xy + 1 = (-\frac{1}{3})x
  3. Convert to Slope-Intercept Form: Convert the point-slope form to slope-intercept form y=mx+by = mx + b to make it easier to compare with the given equations.\newliney+1=(−13)xy + 1 = (-\frac{1}{3})x\newlineSubtract 11 from both sides to isolate yy.\newliney=(−13)x−1y = (-\frac{1}{3})x - 1
  4. Compare with Given Equations: Compare the derived equation y=(−1/3)x−1y = (-1/3)x - 1 with the given equations.\newlineI. y+2=(1/3)(x−3)y + 2 = (1/3)(x - 3) does not match because the slope is positive 1/31/3 and the y-intercept is not −1-1.\newlineII. 2x+6y=−62x + 6y = -6 can be rewritten in slope-intercept form to check if it matches.\newlineDivide the entire equation by 66 to isolate yy.\newline(2x+6y)/6=−6/6(2x + 6y) / 6 = -6 / 6\newline(1/3)x+y=−1(1/3)x + y = -1\newliney=(−1/3)x−1y = (-1/3)x - 1\newlineThis matches the derived equation.
  5. Determine Correct Equation: Determine which of the given equations, if any, represent the line that passes through the points (0,−1)(0,-1) and (−9,2)(-9,2).\newlineEquation I does not represent the line because the slope and yy-intercept do not match.\newlineEquation II does represent the line because, after simplification, it matches the derived equation.

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