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Which of the following equations represents a line that passes through the points (3,-6) and (9,-4) ? I. y=(1)/(3)x-7 II. x-3y=21 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (3,−6) (3,-6) and (9,−4) (9,-4) ?\newlineI. y=13x−7 y=\frac{1}{3} x-7 \newlineII. x−3y=21 x-3 y=21 \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 (3,−6) (3,-6) and (9,−4) (9,-4) ?\newlineI. y=13x−7 y=\frac{1}{3} x-7 \newlineII. x−3y=21 x-3 y=21 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate Slope: First, we need to find the slope of the line that passes through the points (3,−6)(3,-6) and (9,−4)(9,-4). The slope mm is calculated using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  2. Find Y-Intercept: Using the given points (3,−6)(3, -6) and (9,−4)(9, -4), we calculate the slope as follows:\newlinem=−4−(−6)9−3m = \frac{-4 - (-6)}{9 - 3}\newlinem=−4+66m = \frac{-4 + 6}{6}\newlinem=26m = \frac{2}{6}\newlinem=13m = \frac{1}{3}
  3. Slope-Intercept Form: Now that we have the slope, we can use one of the points to find the y-intercept ( extit{b}) of the line. We can use the point (3,−6)(3, -6) and the slope-intercept form equation y=mx+by = mx + b.
  4. Check Equation I: Substitute the values into the equation y=mx+by = mx + b:−6=(13)(3)+b-6 = \left(\frac{1}{3}\right)(3) + b−6=1+b-6 = 1 + bb=−6−1b = -6 - 1b=−7b = -7
  5. Check Equation II: With the slope m=13m = \frac{1}{3} and the y-intercept b=−7b = -7, the equation of the line in slope-intercept form is y=(13)x−7y = \left(\frac{1}{3}\right)x - 7.
  6. Verify Equations: Now let's check if the equation I, y=13x−7y = \frac{1}{3}x - 7, matches the equation we found. It does match, so equation I represents the line that passes through the points (3,−6)(3, -6) and (9,−4)(9, -4).
  7. Verify Equations: Now let's check if the equation I, y=13x−7y = \frac{1}{3}x - 7, matches the equation we found. It does match, so equation I represents the line that passes through the points (3,−6)(3, -6) and (9,−4)(9, -4).Next, we need to check if equation II, x−3y=21x - 3y = 21, also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.
  8. Verify Equations: Now let's check if the equation I, y=13x−7y = \frac{1}{3}x - 7, matches the equation we found. It does match, so equation I represents the line that passes through the points (3,−6)(3, -6) and (9,−4)(9, -4).Next, we need to check if equation II, x−3y=21x - 3y = 21, also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.First, we substitute the point (3,−6)(3, -6) into equation II:\newline3−3(−6)=213 - 3(-6) = 21\newline3+18=213 + 18 = 21\newline21=2121 = 21\newlineThis point satisfies the equation, so we move on to the next point.
  9. Verify Equations: Now let's check if the equation I, y=13x−7y = \frac{1}{3}x - 7, matches the equation we found. It does match, so equation I represents the line that passes through the points (3,−6)(3, -6) and (9,−4)(9, -4).Next, we need to check if equation II, x−3y=21x - 3y = 21, also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.First, we substitute the point (3,−6)(3, -6) into equation II:\newline3−3(−6)=213 - 3(-6) = 21\newline3+18=213 + 18 = 21\newline21=2121 = 21\newlineThis point satisfies the equation, so we move on to the next point.Now, we substitute the point (9,−4)(9, -4) into equation II:\newline9−3(−4)=219 - 3(-4) = 21\newline(3,−6)(3, -6)00\newline21=2121 = 21\newlineThis point also satisfies the equation, so equation II represents the line that passes through the points as well.
  10. Verify Equations: Now let's check if the equation I, y=13x−7y = \frac{1}{3}x - 7, matches the equation we found. It does match, so equation I represents the line that passes through the points (3,−6)(3, -6) and (9,−4)(9, -4).Next, we need to check if equation II, x−3y=21x - 3y = 21, also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.First, we substitute the point (3,−6)(3, -6) into equation II:\newline3−3(−6)=213 - 3(-6) = 21\newline3+18=213 + 18 = 21\newline21=2121 = 21\newlineThis point satisfies the equation, so we move on to the next point.Now, we substitute the point (9,−4)(9, -4) into equation II:\newline9−3(−4)=219 - 3(-4) = 21\newline(3,−6)(3, -6)00\newline21=2121 = 21\newlineThis point also satisfies the equation, so equation II represents the line that passes through the points as well.Since both equations I and II represent the line that passes through the points (3,−6)(3, -6) and (9,−4)(9, -4), the correct answer is ＂I and II.＂

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