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Which of the following equations represents a line that passes through the points (4,-5) and (0,-3) ? I. y=-(1)/(2)x-5 II. 2x+4y=-8 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (4,−5) (4,-5) and (0,−3) (0,-3) ?\newlineI. y=−12x−5 y=-\frac{1}{2} x-5 \newlineII. 2x+4y=−8 2 x+4 y=-8 \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 (4,−5) (4,-5) and (0,−3) (0,-3) ?\newlineI. y=−12x−5 y=-\frac{1}{2} x-5 \newlineII. 2x+4y=−8 2 x+4 y=-8 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate Slope: Calculate the slope of the line passing through the points (4,−5)(4, -5) and (0,−3)(0, -3). The slope (m)(m) is calculated using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Here, (x1,y1)=(4,−5)(x_1, y_1) = (4, -5) and (x2,y2)=(0,−3)(x_2, y_2) = (0, -3). m=−3−(−5)0−4=2−4=−12m = \frac{-3 - (-5)}{0 - 4} = \frac{2}{-4} = -\frac{1}{2}.
  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 (4,−5)(4, -5) and the slope −12-\frac{1}{2}.\newlineThe point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1).\newlineSubstituting the values, we get y−(−5)=−12(x−4)y - (-5) = -\frac{1}{2}(x - 4).
  3. Convert to Slope-Intercept Form: Simplify the equation from point-slope form to slope-intercept form y=mx+by = mx + b.\newliney+5=−12(x−4)y + 5 = -\frac{1}{2}(x - 4)\newliney+5=−12⋅x+2y + 5 = -\frac{1}{2} \cdot x + 2\newliney=−12⋅x+2−5y = -\frac{1}{2} \cdot x + 2 - 5\newliney=−12⋅x−3y = -\frac{1}{2} \cdot x - 3
  4. Check Equation I: Check if equation I, y=−12x−5y = -\frac{1}{2}x - 5, matches the derived equation.\newlineOur derived equation is y=−12×x−3y = -\frac{1}{2} \times x - 3, which does not match equation I.\newlineTherefore, equation I does not represent the line passing through the points (4,−5)(4, -5) and (0,−3)(0, -3).
  5. Check Equation II: Check if equation II, 2x+4y=−82x + 4y = -8, can be simplified to the derived equation.\newlineFirst, we convert equation II to slope-intercept form.\newline2x+4y=−82x + 4y = -8\newline4y=−2x−84y = -2x - 8\newliney=(−2x−8)/4y = (-2x - 8) / 4\newliney=−12∗x−2y = -\frac{1}{2} * x - 2\newlineThis equation also does not match our derived equation y=−12∗x−3y = -\frac{1}{2} * x - 3.\newlineTherefore, equation II does not represent the line passing through the points (4,−5)(4, -5) and (0,−3)(0, -3).

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