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### Two points have coordinates (1,-6) and (-2,3). Find the equation of the line which joins them, and their midpoint.

To find the line's equation, you need the gradient and the y-intercept. The gradient is found using:

(y2-y1)/(x2-x1)

Or simply (change in y direction)/(change in x direction)

In this case, the gradient =(3+6)/(-2-1)=9/-3=-3

and so the equation of the line is in the form

y=-3x+c

We now need to find the y-intercept, or 'c' in the equation above.

All points on this line will follow this equation. This includes the two points in the question, so we can sub in the x and y values and rearrange to find c. We can choose either of the points. Lets start with (1,-6)

x=1, y=-6

y=-3x + c

-6=-3*1 +c

-6=-3 +c

c=-3

Therefore:

y=-3x-3

We can check this by using the other coordinate:

(-2,3)

x=-2, y=3

3=-3*-2  -3

3=6-3

3=3

As for finding the midpoint of the two points, we just need to find middle of the 2 x-coordinates and the middle of the 2 y-coordinates (need some diagrams here)

To do this, we can just find the averages.

x - coordinates: 1 and -2

x midpoint = (1+ -2)/2 = -0.5

y - coordinates: -6 and 3

y midpoint = (-6+3)/2 =-1.5

So our midpoint coordinate is (-0.5, -1.5)

2 years ago

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