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Year 8 Maths Chapter 7 - Linear Relationships 1

You will want to refer to Year 9 Maths Chapter 4 for more advanced knowledge of linear relationships. This section is more advanced. For quadratics take a look at Year 9 Maths Chapter 8 and Year 9 Maths Chapter 10

Basic terms and information:

  • A number plane (or Cartesian plane) includes a y-axis and an x-axis, intersecting at right angles, forming four quadrants.
  • Anti-clockwise from top-right: Quadrant 1 - Quadrant 4.
  • Point on a number plane has coordinates: (x, y), where x is number of horizontal units from origin (x-coordinate) and vice versa (vertical units, y-coordinate).
  • The point (0, 0) is known as the origin.
  • A rule or a formula is an equation connecting two or more variables.
  • A straight line forms a linear rule.
  • Linear rules are often written with y as the subject. For example, y = 3x - 2

You can use a table of values to find a rule. The rule must be true for every pair of coordinates, substituted into a table or graph.

For example:

y = ☐x + ☐

To find a rule, use the formula below.

y = mx + b

where m is the gradient and b is the y-intercept. This is known as gradient-intercept form.

  • The gradient is rise over run.
  • On a graph, it is the steepness of a line, measured by a specific height measurement, over a specific length measurement.
  • For example, for a gradient to be 2, on a graph the line would be increasing in height, and for every 5 units on the x axis, there would be a growth of 10 units.
  • In a table of values, it is the difference in the y-axis between every consecutive number on the x-axis
  • The y-intercept is the constant value when x = 0.
  • The x-intercept is the constant value where y = 0.

Gradient-intercept form

A linear equation has a solution, where a pair of values, representing x-coordinates and y-coordinates, solve an equation when substituted into their relative pronumerals.

Where there are two straight lines crossing over, there is a point of intersection (the point where the lines cross). In this case, the point of intersection is the pair of values that satisfies both rules.

Point of intersection

Non-linear relationships are graphs or rules that are not linear (form a curve, such as a parabola).

An example of a non-linear curve:

y = x^2

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