# 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.*

#### Table of Contents

- Intercepts
- Gradient
- Gradient-intercept form
- General Form
- Midpoints
- Distance formula
- Direct proportion/direct variance
- Parallel lines and perpendicular lines:

## Intercepts

x-intercept = `y = 0`

y-intercept = `x = 0`

***

## Gradient

The gradient is a measure of slope.

```
rise
Gradient = ––––––
run
```

Be careful when the gradient is negative.

## Gradient-intercept form

```
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
```