# Year 9 Maths Chapter 4 – Linear Relationships

You will want to refer to Year 8 Maths Chapter 7 for basic linear relationships. This section is more advanced, and omits basic concepts.

## Intercepts

x-intercept = `y = 0` y-intercept = `x = 0` ***

The gradient is a measure of slope.

``````            rise
run
``````

Be careful when the gradient is negative.

``````y = mx + b
``````

where `m` is the gradient, and `b` is the y-intercept. ***

## General Form

``````ax + by = c
``````

or

``````ax + by + c = 0
``````

where a, b and c are constants. It is typically the result of an expanded FOIL. ***

## Midpoints

The midpoint is the halfway point between two end points. To find the midpoint, use the midpoint formula.

``````    ┌ x1 + x2   y1 + y2 ┐
M = | ––––––– , ––––––– |
└    2         2    ┘
``````

The `x` coordinate is the average of the two `x` coordinates The `y` coordinate is the average of the two `y` coordinates

## Distance formula

To find a length of a line, use Pythag.

Alternatively, use the Distance Formula.

``````     ___________________________
d = √ (x2 - x1)^2 + (y2 - y1)^2
``````

#### Example

``````     _______________________________
d = √ ((-7) - (-3))^2 + ((-4)-2)^2
_________________
d = √ (-4)^2 + (-6)^2

d = √52
d = 2√13
d = 7.21 units (2dp)
``````

## Direct proportion/direct variance

If b is directly proportional to a:

``````b = ka
``````

where k is a constant.

Also known as:

``````y = mx
``````

## Parallel lines and perpendicular lines:

If two lines are parallel they have the same gradient. If two lines are perpendicular (at right angles) then they are reciprocals of each other.

``````Let the gradients of two perpendicular lines be m1 and m2.

m1 x m2 = -1
1
m2 = –––
m1

Hence, m2 is equal to the reciprocal of m1
``````

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