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Which of the following equations represents a line that passes through the points (8,-2) and (4,3) ? I. y=-(5)/(4)x+10 II. 5x+4y=32 Neither I only II only I and II

Which of the following equations represents a line that passes through the points (8,−2) (8,-2) and (4,3) (4,3) ?\newlineI. y=−54x+10 y=-\frac{5}{4} x+10 \newlineII. 5x+4y=32 5 x+4 y=32 \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 (8,−2) (8,-2) and (4,3) (4,3) ?\newlineI. y=−54x+10 y=-\frac{5}{4} x+10 \newlineII. 5x+4y=32 5 x+4 y=32 \newlineNeither\newlineI only\newlineII only\newlineI and II
  1. Calculate slope between two points: Calculate the slope of the line passing through the points (8,−2)(8,-2) and (4,3)(4,3). The slope (m)(m) is given by the formula m=(y2−y1)(x2−x1)m = \frac{(y_2 - y_1)}{(x_2 - x_1)}. Using the points (8,−2)(8,-2) (x1,y1)(x_1,y_1) and (4,3)(4,3) (x2,y2)(x_2,y_2), we get: m=(3−(−2))(4−8)m = \frac{(3 - (-2))}{(4 - 8)} m=(3+2)(4−8)m = \frac{(3 + 2)}{(4 - 8)} (4,3)(4,3)00 (4,3)(4,3)11
  2. Check slope in equation I: Check if equation I, y=−54x+10y = -\frac{5}{4}x + 10, has the correct slope.\newlineThe slope of equation I is −54-\frac{5}{4}, which matches the slope we calculated in Step 11.
  3. Substitute point into equation I: Substitute one of the points into equation I to see if it satisfies the equation.\newlineLet's use the point (8,−2)(8,-2).\newliney=−54x+10y = -\frac{5}{4}x + 10\newline−2=−54(8)+10-2 = -\frac{5}{4}(8) + 10\newline−2=−10+10-2 = -10 + 10\newline−2=0-2 = 0\newlineThis is not true, so equation I does not pass through the point (8,−2)(8,-2).
  4. Write line equation using slope: Write the general form of the line equation using the slope and one of the points.\newlineUsing the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1), with m=−54m = -\frac{5}{4} and point (8,−2)(8,-2), we get:\newliney−(−2)=−54(x−8)y - (-2) = -\frac{5}{4}(x - 8)\newliney+2=−54x+10y + 2 = -\frac{5}{4}x + 10\newliney=−54x+10−2y = -\frac{5}{4}x + 10 - 2\newliney=−54x+8y = -\frac{5}{4}x + 8\newlineThis is the equation of the line in slope-intercept form.
  5. Check equation II for same line: Check if equation II, 5x+4y=325x + 4y = 32, represents the same line.\newlineWe can convert equation II to slope-intercept form to compare the slopes and y-intercepts.\newline5x+4y=325x + 4y = 32\newline4y=−5x+324y = -5x + 32\newliney=−54x+8y = -\frac{5}{4}x + 8\newlineThis matches the equation we derived in Step 44, so equation II has both the correct slope and y-intercept.
  6. Verify equation II with point (8,−2)(8,-2): Verify that equation II passes through both points by substituting them into the equation.\newlineFirst, use the point (8,−2)(8,-2):\newline5(8)+4(−2)=325(8) + 4(-2) = 32\newline40−8=3240 - 8 = 32\newline32=3232 = 32\newlineThis is true, so the point (8,−2)(8,-2) lies on the line represented by equation II.
  7. Verify equation II with point (4,3)(4,3): Now, use the point (4,3)(4,3) to verify equation II:\newline5(4)+4(3)=325(4) + 4(3) = 32\newline20+12=3220 + 12 = 32\newline32=3232 = 32\newlineThis is true, so the point (4,3)(4,3) also lies on the line represented by equation II.

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