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Which of the following equations represents a line that passes through the points (-5,4) and (-10,3) ? I. x-5y=-25 II. y-4=(1)/(5)(x+5) Neither I only II only I and II

Which of the following equations represents a line that passes through the points (−5,4) (-5,4) and (−10,3) (-10,3) ?\newlineI. x−5y=−25 x-5 y=-25 \newlineII. y−4=15(x+5) y-4=\frac{1}{5}(x+5) \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 (−5,4) (-5,4) and (−10,3) (-10,3) ?\newlineI. x−5y=−25 x-5 y=-25 \newlineII. y−4=15(x+5) y-4=\frac{1}{5}(x+5) \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate slope: Calculate the slope of the line passing through the points (−5,4)(-5,4) and (−10,3)(-10,3). The slope mm is given by the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Using the points (−5,4)(-5,4) and (−10,3)(-10,3), we get m=3−4−10+5=−1−5=15m = \frac{3 - 4}{-10 + 5} = \frac{-1}{-5} = \frac{1}{5}.
  2. Write equation: Use the slope and one of the points to write the equation of the line in point-slope form.\newlineThe point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1).\newlineUsing the slope 15\frac{1}{5} and the point (−5,4)(-5,4), we get y−4=(15)(x−(−5))y - 4 = \left(\frac{1}{5}\right)(x - (-5)) or y−4=15(x+5)y - 4 = \frac{1}{5}(x + 5).
  3. Check equation I: Check if equation I, x−5y=−25x - 5y = -25, represents the line with the slope 15\frac{1}{5}. To do this, we can rearrange the equation into slope-intercept form (y=mx+by = mx + b) and compare the slope. x−5y=−25x - 5y = -25 can be rewritten as 5y=x+255y = x + 25, and then y=(15)x+5y = \left(\frac{1}{5}\right)x + 5. The slope of this line is 15\frac{1}{5}, which matches the slope we calculated in Step 11.
  4. Check equation II: Check if equation II, y−4=(15)(x+5)y - 4 = \left(\frac{1}{5}\right)(x + 5), represents the line with the slope 15\frac{1}{5}. This equation is already in point-slope form, and it matches the equation we derived in Step 22. Therefore, equation II also represents the line passing through the points (−5,4)(-5,4) and (−10,3)(-10,3).
  5. Determine final answer: Determine the final answer based on the checks in Steps 33 and 44. Since both equations II and IIII have the correct slope and pass through one of the given points, they both represent the line passing through the points (−5,4)(-5,4) and (−10,3)(-10,3).

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