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Which of the following equations represents a line that passes through the points (-4,-6) and (2,0) ? I. y+6=(x+9) II. 5x-5y=15 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (−4,−6) (-4,-6) and (2,0) (2,0) ?\newlineI. y+6=(x+9) y+6=(x+9) \newlineII. 5x−5y=15 5 x-5 y=15 \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,−6) (-4,-6) and (2,0) (2,0) ?\newlineI. y+6=(x+9) y+6=(x+9) \newlineII. 5x−5y=15 5 x-5 y=15 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate Slope: Find the slope of the line passing through the points (−4,−6)(-4,-6) and (2,0)(2,0). The slope mm is calculated using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. m=0−(−6)2−(−4)m = \frac{0 - (-6)}{2 - (-4)} m=66m = \frac{6}{6} m=1m = 1
  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 (2,0)(2,0) and the slope m=1m = 1.\newlineThe point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1).\newliney−0=1(x−2)y - 0 = 1(x - 2)\newliney=x−2y = x - 2
  3. Check Equation I: Check if equation I, y+6=(x+9)y + 6 = (x + 9), represents the line with the slope found in Step 11.\newlineRewrite equation I in slope-intercept form (y=mx+by = mx + b).\newliney=−x−9+6y = -x - 9 + 6\newliney=−x−3y = -x - 3\newlineThe slope of this line is −1-1, which is not equal to the slope we found in Step 11 (m=1m = 1).\newlineTherefore, equation I does not represent the line passing through the points (−4,−6)(-4,-6) and (2,0)(2,0).
  4. Check Equation II: Check if equation II, 5x−5y=155x - 5y = 15, represents the line with the slope found in Step 11.\newlineRewrite equation II in slope-intercept form (y=mx+by = mx + b).\newline5y=5x−155y = 5x - 15\newliney=x−3y = x - 3\newlineThe slope of this line is 11, which is equal to the slope we found in Step 11 (m=1m = 1).\newlineNow, we need to check if the line passes through one of the given points.
  5. Verify Point in Equation II: Substitute the point (2,0)(2,0) into equation II to verify if it satisfies the equation.\newliney=x−3y = x - 3\newline0=2−30 = 2 - 3\newline0=−10 = -1\newlineThis is not true, so the point (2,0)(2,0) does not lie on the line represented by equation II.\newlineTherefore, equation II also does not represent the line passing through the points (−4,−6)(-4,-6) and (2,0)(2,0).

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