Linear Equations

New Zealand

Level 7 - NCEA Level 2

Lesson

Gradient is the steepness or slope of a line. We've already learnt how to calculate gradient but let's just refresh ourselves on a couple of key points to start.

The Gradient Formula

$m=\frac{y_2-y_1}{x_2-x_1}$`m`=`y`2−`y`1`x`2−`x`1

Other handy points to remember:

- The gradient formula can be thought of generally as: $\text{Gradient }=\frac{\text{rise }}{\text{run }}$Gradient =rise run
- Gradient of Vertical Line is undefined
- Gradient of Horizontal Line $\text{= 0}$= 0

Even though we are used to finding the gradient of a line from two given pairs of coordinates, we may also need to reverse the process. Sometimes, we may be asked to find a pair of coordinates when we're given the gradient and a point.

Check out the examples to see the different ways we can work with the gradient formula.

Consider the following ramp:

a) What is the gradient of this skateboard ramp if it measures $0.9$0.9 metres high and $1$1 metre across?

Think: What is the rise and run of this ramp?

Do:

$\text{Gradient }$Gradient | $=$= | $\frac{\text{Rise }}{\text{Run }}$Rise Run |

$=$= | $\frac{0.9}{1}$0.91 | |

$=$= | $0.9$0.9 |

b) It can only be used as a 'beginner’s ramp' if for every $1$1 metre horizontal run, it has a rise of at most $0.4$0.4 metres. Can it be used as a 'beginner’s ramp'?

Think: What is the maximum gradient of a beginners' ramp? Is the ramp steeper than this?

Do: A beginner's ramp needs to have a gradient of $\frac{0.4}{1}$0.41 or $0.4$0.4. Since the ramp has a steeper gradient of $0.9$0.9, it *cannot* be used as a beginners' ramp.

A line passes through the points $\left(11,c\right)$(11,`c`) and $\left(-20,16\right)$(−20,16) and has a gradient of $-\frac{4}{7}$−47.

Find the value of $c$`c`.

Apply co-ordinate geometry techniques to points and lines

Apply co-ordinate geometry methods in solving problems