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Which of the following equations represents a line that passes through the points (6,4) and (-3,1) ? I. y-4=-(1)/(3)(x-6) II. 2x-6y=-12 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (6,4) (6,4) and (−3,1) (-3,1) ?\newlineI. y−4=−13(x−6) y-4=-\frac{1}{3}(x-6) \newlineII. 2x−6y=−12 2 x-6 y=-12 \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 (6,4) (6,4) and (−3,1) (-3,1) ?\newlineI. y−4=−13(x−6) y-4=-\frac{1}{3}(x-6) \newlineII. 2x−6y=−12 2 x-6 y=-12 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate slope: Calculate the slope of the line passing through the points (6,4)(6,4) and (−3,1)(-3,1). The slope mm is given by the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Using the points (6,4)(6,4) and (−3,1)(-3,1), we get m=1−4−3−6m = \frac{1 - 4}{-3 - 6}. m=−3−9m = \frac{-3}{-9}. m=13m = \frac{1}{3}.
  2. Point-slope form: Use the point-slope form of the equation of a line to write the equation for the line with the slope from Step 11 that passes through one of the given points, say (6,4)(6,4).\newlineThe point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1).\newlineUsing the slope m=13m = \frac{1}{3} and point (6,4)(6,4), we get y−4=13(x−6)y - 4 = \frac{1}{3}(x - 6).
  3. Check equation I: Check if equation I, y−4=−13(x−6)y - 4 = -\frac{1}{3}(x - 6), represents the line with the slope from Step 11.\newlineThe slope of equation I is −13-\frac{1}{3}, which is the negative of the slope we calculated.\newlineTherefore, equation I does not represent the line passing through the points (6,4)(6,4) and (−3,1)(-3,1).
  4. Convert to slope-intercept form: Convert the point-slope form of the equation from Step 22 to the slope-intercept form y=mx+by = mx + b to compare with equation II.\newliney−4=(13)(x−6)y - 4 = \left(\frac{1}{3}\right)(x - 6)\newliney=(13)x−(13)⋅6+4y = \left(\frac{1}{3}\right)x - \left(\frac{1}{3}\right)\cdot6 + 4\newliney=(13)x−2+4y = \left(\frac{1}{3}\right)x - 2 + 4\newliney=(13)x+2y = \left(\frac{1}{3}\right)x + 2
  5. Convert equation II: Convert equation II, 2x−6y=−122x - 6y = -12, to the slope-intercept form to find its slope and y-intercept.\newlineDivide the entire equation by −6-6 to solve for y.\newline−6y=−2x+12-6y = -2x + 12\newliney=(13)x−2y = \left(\frac{1}{3}\right)x - 2
  6. Compare equations: Compare the slope-intercept form of the line from Step 44 with equation II.\newlineBoth equations have the same slope, 13\frac{1}{3}, and the same yy-intercept, −2-2.\newlineTherefore, equation II represents the line passing through the points (6,4)(6,4) and (−3,1)(-3,1).
  7. Determine correct answer: Determine the correct answer based on the analysis of equations I and II.\newlineEquation I does not represent the line because its slope is the negative of the correct slope.\newlineEquation II does represent the line because it has the correct slope and yy-intercept.\newlineThe correct answer is ＂II only＂.

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