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Complete the point-slope equation of the line through (1,−1)(1,-1) and (5,2)(5,2) Use exact numbers.

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Full solution

Q. Complete the point-slope equation of the line through (1,−1)(1,-1) and (5,2)(5,2) Use exact numbers.
  1. Calculate slope: Calculate the slope mm of the line using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Using the points (1,−1)(1, -1) and (5,2)(5, 2), we have: m=2−(−1)5−1m = \frac{2 - (-1)}{5 - 1} m=2+14m = \frac{2 + 1}{4} m=34m = \frac{3}{4}
  2. Use point-slope form: Use the point-slope form of the equation of a line, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line.\newlineWe can use either of the given points for (x1,y1)(x_1, y_1). Let's use the point (1,−1)(1, -1).
  3. Substitute slope and point: Substitute the slope mm and the point (1,−1)(1, -1) into the point-slope equation.\newliney−(−1)=(34)(x−1)y - (-1) = \left(\frac{3}{4}\right)(x - 1)\newliney+1=(34)(x−1)y + 1 = \left(\frac{3}{4}\right)(x - 1)
  4. Final equation: This is the point-slope form of the equation of the line. No further simplification is required since the problem asks for the point-slope form and not the slope-intercept form.

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