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Calculate the slope m.

Mathematics

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Here are the equations for each pair of points:

  • a) Points: (2,7)(2, 7) and (4,−2)(4, -2)

  1. Calculate the slope mm.

    m=y2−y1x2−x1=−2−74−2=−92m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 7}{4 - 2} = \frac{-9}{2}

  2. Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (2,7)(2, 7).

    y−7=−92(x−2)y - 7 = -\frac{9}{2}(x - 2)

  3. Simplify the equation.

    y−7=−92x+9y - 7 = -\frac{9}{2}x + 9 y=−92x+9+7y = -\frac{9}{2}x + 9 + 7 y=−92x+16y = -\frac{9}{2}x + 16

    The equation of the line is y=−92x+16\boxed{y = -\frac{9}{2}x + 16}.

    • b) Points: (−1,5)(-1, 5) and (4,−2)(4, -2)

  4. Calculate the slope mm.

    m=y2−y1x2−x1=−2−54−(−1)=−75m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 5}{4 - (-1)} = \frac{-7}{5}

  5. Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (−1,5)(-1, 5).

    y−5=−75(x−(−1))y - 5 = -\frac{7}{5}(x - (-1)) y−5=−75(x+1)y - 5 = -\frac{7}{5}(x + 1)

  6. Simplify the equation.

    y−5=−75x−75y - 5 = -\frac{7}{5}x - \frac{7}{5} y=−75x−75+5y = -\frac{7}{5}x - \frac{7}{5} + 5 y=−75x−75+255y = -\frac{7}{5}x - \frac{7}{5} + \frac{25}{5} y=−75x+185y = -\frac{7}{5}x + \frac{18}{5}

    The equation of the line is y=−75x+185\boxed{y = -\frac{7}{5}x + \frac{18}{5}}.

    • c) Points: (−2,3)(-2, 3) and (4,5)(4, 5)

  7. Calculate the slope mm.

    m=y2−y1x2−x1=5−34−(−2)=26=13m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}

  8. Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (−2,3)(-2, 3).

    y−3=13(x−(−2))y - 3 = \frac{1}{3}(x - (-2)) y−3=13(x+2)y - 3 = \frac{1}{3}(x + 2)

  9. Simplify the equation.

    y−3=13x+23y - 3 = \frac{1}{3}x + \frac{2}{3} y=13x+23+3y = \frac{1}{3}x + \frac{2}{3} + 3 y=13x+23+93y = \frac{1}{3}x + \frac{2}{3} + \frac{9}{3} y=13x+113y = \frac{1}{3}x + \frac{11}{3}

    The equation of the line is y=13x+113\boxed{y = \frac{1}{3}x + \frac{11}{3}}.

    • d) Points: (−1,−8)(-1, -8) and (3,−2)(3, -2)

  10. Calculate the slope mm.

    m=y2−y1x2−x1=−2−(−8)3−(−1)=−2+83+1=64=32m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-8)}{3 - (-1)} = \frac{-2 + 8}{3 + 1} = \frac{6}{4} = \frac{3}{2}

  11. Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (−1,−8)(-1, -8).

    y−(−8)=32(x−(−1))y - (-8) = \frac{3}{2}(x - (-1)) y+8=32(x+1)y + 8 = \frac{3}{2}(x + 1)

  12. Simplify the equation.

    y+8=32x+32y + 8 = \frac{3}{2}x + \frac{3}{2} y=32x+32−8y = \frac{3}{2}x + \frac{3}{2} - 8 y=32x+32−162y = \frac{3}{2}x + \frac{3}{2} - \frac{16}{2} y=32x−132y = \frac{3}{2}x - \frac{13}{2}

    The equation of the line is y=32x−132\boxed{y = \frac{3}{2}x - \frac{13}{2}}.

    • e) Points: (3,7)(3, 7) and (8,4)(8, 4)

  13. Calculate the slope mm.

    m=y2−y1x2−x1=4−78−3=−35m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 7}{8 - 3} = \frac{-3}{5}

  14. Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (3,7)(3, 7).

    y−7=−35(x−3)y - 7 = -\frac{3}{5}(x - 3)

  15. Simplify the equation.

    y−7=−35x+95y - 7 = -\frac{3}{5}x + \frac{9}{5} y=−35x+95+7y = -\frac{3}{5}x + \frac{9}{5} + 7 y=−35x+95+355y = -\frac{3}{5}x + \frac{9}{5} + \frac{35}{5} y=−35x+445y = -\frac{3}{5}x + \frac{44}{5}

    The equation of the line is y=−35x+445\boxed{y = -\frac{3}{5}x + \frac{44}{5}}.

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