Year 9 Maths Chapter 4 – Linear Relationships

Complete

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
  • Example
• 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