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Which of the following equations represents a line that passes through the points (5,-7) and (0,-8) ? I. x+5y=40 II. y=(1)/(5)x-10 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (5,−7) (5,-7) and (0,−8) (0,-8) ?\newlineI. x+5y=40 x+5 y=40 \newlineII. y=15x−10 y=\frac{1}{5} x-10 \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,−7) (5,-7) and (0,−8) (0,-8) ?\newlineI. x+5y=40 x+5 y=40 \newlineII. y=15x−10 y=\frac{1}{5} x-10 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate Slope: Find the slope of the line passing through the points (5,−7)(5,-7) and (0,−8)(0,-8). The slope mm is calculated using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Here, (x1,y1)=(5,−7)(x_1, y_1) = (5, -7) and (x2,y2)=(0,−8)(x_2, y_2) = (0, -8). m=−8−(−7)0−5=−8+7−5=−1−5=15m = \frac{-8 - (-7)}{0 - 5} = \frac{-8 + 7}{-5} = \frac{-1}{-5} = \frac{1}{5}.
  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 (5,−7)(5, -7) and the slope 15\frac{1}{5}.\newlineThe point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1).\newlineSubstituting the values, we get y−(−7)=(15)(x−5)y - (-7) = \left(\frac{1}{5}\right)(x - 5).
  3. Convert to Slope-Intercept Form: Simplify the equation to get it into slope-intercept form y=mx+by = mx + b.\newliney+7=15x−15⋅5y + 7 = \frac{1}{5}x - \frac{1}{5}\cdot5\newliney+7=15x−1y + 7 = \frac{1}{5}x - 1\newlineSubtract 77 from both sides to isolate yy.\newliney=15x−1−7y = \frac{1}{5}x - 1 - 7\newliney=15x−8y = \frac{1}{5}x - 8
  4. Check Given Equations: Check if the equation y=15x−8y = \frac{1}{5}x - 8 matches any of the given equations.\newlineEquation I: x+5y=40x + 5y = 40\newlineEquation II: y=15x−10y = \frac{1}{5}x - 10\newlineThe equation we derived, y=15x−8y = \frac{1}{5}x - 8, matches neither Equation I nor Equation II exactly.
  5. Verify Equation I: Verify if Equation I or Equation II passes through both given points.\newlineFor Equation I: x+5y=40x + 5y = 40\newlineSubstitute point (5,−7)(5, -7) into the equation: 5+5(−7)=5−35=−305 + 5(-7) = 5 - 35 = -30, which does not equal 4040.\newlineSubstitute point (0,−8)(0, -8) into the equation: 0+5(−8)=−400 + 5(-8) = -40, which does equal 4040.\newlineSo, Equation I does not pass through both points.
  6. Verify Equation II: Verify if Equation II passes through both given points.\newlineFor Equation II: y=15x−10y = \frac{1}{5}x - 10\newlineSubstitute point (5,−7)(5, -7) into the equation: −7=(15)∗5−10=1−10=−9-7 = \left(\frac{1}{5}\right)*5 - 10 = 1 - 10 = -9, which does not equal −7-7.\newlineSubstitute point (0,−8)(0, -8) into the equation: −8=(15)∗0−10=0−10=−10-8 = \left(\frac{1}{5}\right)*0 - 10 = 0 - 10 = -10, which does not equal −8-8.\newlineSo, Equation II does not pass through both points.

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