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Given a graph with points A and B, determine the coordinates of A and B, the gradient of line AB, and the equation of line AB in the form y = mx + c.

Mathematics

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Answers

3 parts
a
(4, 3)
(6, -5)
b
-4
c

y=−4x+19y = -4x + 19

Step-by-step solution

8 steps

Part a· 2 steps

Write the coordinates of points A and B.

  1. Identify the coordinates of point A from the graph.

    Point A is located at x=4x=4 and y=3y=3.
    Coordinates of A: (4,3)\boxed{(4, 3)}

  2. Identify the coordinates of point B from the graph.

    Point B is located at x=6x=6 and y=−5y=-5.
    Coordinates of B: (6,−5)\boxed{(6, -5)}

Part b-4

Determine the gradient of line AB.

  1. Use the coordinates of points A and B.

    Let A=(x1,y1)=(4,3)A = (x_1, y_1) = (4, 3)
    Let B=(x2,y2)=(6,−5)B = (x_2, y_2) = (6, -5)

  2. Apply the gradient formula.

    The gradient mm is given by the formula:

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
  3. Substitute the coordinates into the formula.

    m=−5−36−4m = \frac{-5 - 3}{6 - 4} m=−82m = \frac{-8}{2} m=−4m = -4

    The gradient of line AB is −4\boxed{-4}.

Part c

y=−4x+19y = -4x + 19

Determine the equation of line AB in the form y=mx+cy = mx + c.

  1. Use the gradient m=−4m = -4 (from part b) and one of the points, for example, point A (4,3)(4, 3).

    The equation of a line can be written in the point-slope form: y−y1=m(x−x1)y - y_1 = m(x - x_1).

  2. Substitute the gradient and the coordinates of point A into the point-slope form.

    y−3=−4(x−4)y - 3 = -4(x - 4)
  3. Simplify the equation to the form y=mx+cy = mx + c.

    y−3=−4x+(−4)(−4)y - 3 = -4x + (-4)(-4) y−3=−4x+16y - 3 = -4x + 16 y=−4x+16+3y = -4x + 16 + 3 y=−4x+19y = -4x + 19

    The equation of line AB is y=−4x+19\boxed{y = -4x + 19}.

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