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Gradient

Gradient = up divided by across The gradient of a line with points (-2,0) and (0,2), 2 divided by2 = 1 This is a 45 degree line

A = (x1,y1) B = (x2,y2) The gradient of AB is y1 - y2 divided by x1 -x2 or y2 - y1 divided by x2 - x1

dy/dx means the gradient of a line

Perpendicular is the same as normal The normal of dy/dx 2 is -1/2

The normal of dy/dx x/y is -y/x x = 2 y = 1

Two lines with equal gradients are parallel

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Midpoints

Point A is (-1,2) Point B is (3,-4) M is the midpoint of the line joining A and B.

• The x-coordinate of M is (-1 + 3) divided by 2 = 1• The y-coordinate of M is (2 + -4) divided by 2 = -1 The midpoint is (1,-1)

A = (x1,y1) B = (x2,y2) The midpoint of AB is ((x1 + x2) divided by 2, (y1 + y2) divided by 2)

Point A is (-5,3) Point B is (4,0) Find the length of AB

• The x-axis = 9 The y-axis = 3 9 squared = 81 3 squared = 9 81 + 9 = 90 root 90= 9.5 Line AB = 9.5

A = (x1,y1) B = (x2,y2) The midpoint of AB is root(x2 - x1) squared + (y2 - y1) squared

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Equation of a Line

If we know one point on a line and the gradient the formula y - y1 = m(x - x1) is used to find theequation or the y-intercept

The gradient is 2 and the point is (1,3) y - 3 = 2(x -1) y - 3 = 2x - 2 y = 2x + 1

If we don't know the gradient we can still use the formula with two points as well

The points are (2,4) and (7,3) m = (3-4)/(7-2) m = -1/5 y - 4 = -1/5(x - 2) 5y - 20 = -x + 2 5y = -x + 22 it can be writen x + 5y = 22

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y = mx + c

m is the gradient y = 2x + c has a gradient of 2

c is the y-intercept y = mx + 4 crosses the y axis at (0,4)

First rearrange the equation to write y = mx + c make sure the y is an integer (whole number)

If x is a fraction the denominator is the across value while the numerator is the up value (n/d)

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Intersecting Lines

Unless two lines are parallel they will intersect

First make both lines equal each other through y or x, then find the value of x or y

x + y = 6 2x - y = 9 y = -x + 6 y = 2x - 9 2x - 9 = -x + 6 3x = 15 x = 5

Then substitute the found number into either equation

5 + y = 6 y = 1 the point where the lines intersect is (5,1)