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Which of the following equations represents a line that passes through the points (-4,3) and (7,14) ? I. y-3=(x+4) II. y=x+8 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (−4,3) (-4,3) and (7,14) (7,14) ?\newlineI. y−3=(x+4) y-3=(x+4) \newlineII. y=x+8 y=x+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 (−4,3) (-4,3) and (7,14) (7,14) ?\newlineI. y−3=(x+4) y-3=(x+4) \newlineII. y=x+8 y=x+8 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate Slope: To find the equation of the line that passes through two points, we can use the point-slope form of a line, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the line and (x1,y1)(x_1, y_1) is one of the points it passes through. First, we need to calculate the slope (mm) using the two given points (−4,3)(-4,3) and (7,14)(7,14). The slope mm is calculated by the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Let's calculate the slope: m=14−37−(−4)m = \frac{14 - 3}{7 - (-4)} m=1111m = \frac{11}{11} mm00
  2. Write Equation: Now that we have the slope, we can use either point to write the equation of the line in point-slope form. Let's use the point (−4,3)(-4,3).\newlineThe equation will be y−3=1(x−(−4))y - 3 = 1(x - (-4)), which simplifies to y−3=1(x+4)y - 3 = 1(x + 4).
  3. Check Equation: Let's check if the equation y−3=1(x+4)y - 3 = 1(x + 4) is equivalent to any of the given equations.\newlineFirst, we simplify y−3=1(x+4)y - 3 = 1(x + 4) to y=x+4+3y = x + 4 + 3, which gives us y=x+7y = x + 7.\newlineThis equation is not equivalent to either of the given equations (I. y−3=(x+4)y - 3 = (x + 4) or II. y=x+8y = x + 8).
  4. Check Equation: Now let's check the second given equation, y=x+8y = x + 8, to see if it represents the line that passes through the points (−4,3)(-4,3) and (7,14)(7,14). We can plug in the xx-values of the points into the equation to see if we get the corresponding yy-values. For the point (−4,3)(-4,3), if we plug in x=−4x = -4, we get y=−4+8y = -4 + 8, which simplifies to y=4y = 4. This does not match the yy-value of the point (−4,3)(-4,3), which is (−4,3)(-4,3)11.
  5. Check Equation: Next, we check the point (7,14)(7,14) with the equation y=x+8y = x + 8. If we plug in x=7x = 7, we get y=7+8y = 7 + 8, which simplifies to y=15y = 15. This does not match the y-value of the point (7,14)(7,14), which is 1414.
  6. Final Answer: Since neither equation I y−3=(x+4)y - 3 = (x + 4) nor equation II y=x+8y = x + 8 matches the equation we derived y=x+7y = x + 7 and neither produces the correct yy-values when we plug in the xx-values of the given points, the correct answer is ＂Neither.＂

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