Linear Equations II

Hong Kong

Stage 1 - Stage 3

Lesson

Some lines have increasing slopes, like these:

And some have decreasing slopes, like these:

This applet will let you create lines with positive and negative gradients:

The slope of a line is a measure of how steep it is. In mathematics we call this the **gradient**.

A gradient is a single value that describes:

- if a line is increasing (has positive gradient)
- if a line is decreasing (has negative gradient)
- how far up or down the line moves (how the $y$
`y`-value changes) with every step to the right (for every $1$1 unit increase in the $x$`x`-value)

Take a look at this line, where the horizontal and vertical steps are highlighted:

We call the horizontal measurement the **run** and the vertical measurement the **rise**. For this line, a run of $1$1 means a rise of $2$2, so the line has gradient $2$2.

Sometimes it is difficult to measure how far the line goes up or down (how much the $y$`y` value changes) in $1$1 horizontal unit, especially if the line doesn't line up with the grid points on the $xy$`x``y`-plane. In this case we calculate the gradient by using a formula:

$\text{gradient }=\frac{\text{rise }}{\text{run }}$gradient =rise run

The rise and run are calculated from two known points on the line.

You can find the rise and run of a line by drawing a right triangle created by any two points on the line. The line itself forms the hypotenuse.

This line has a gradient of $\frac{\text{rise }}{\text{run }}=\frac{4}{3}$rise run =43

In this case, the gradient is positive because, over the $3$3 unit increase in the $x$`x`-values, the $y$`y`-value has **increased**. If the $y$`y`-value **decreased** as the $x$`x`-value increases, the gradient would be negative.

This applet allows you to see the rise and run between two points on a line of your choosing:

If you have a pair of coordinates, such as $A=\left(3,6\right)$`A`=(3,6) and $B=\left(7,-2\right)$`B`=(7,−2), we can find the gradient of the line between these points using the same formula. It is a good idea to draw a quick sketch of the points, which helps us quickly identify what the line will look like:

Already we can tell that the gradient will be negative, since the line moves downward as we go from left to right.

The rise is the difference in the $y$`y`-values of the points. We take the $y$`y`-value of the rightmost point and subtract the $y$`y`-value of the leftmost point to describe the change in vertical distance from $A$`A` to $B$`B`:

$\text{rise}=-2-6=-8$rise=−2−6=−8.

The run is the difference in the $x$`x`-values of the points. We take the $x$`x`-value of the rightmost point and subtract the $x$`x`-value of the leftmost point to describe the change in horizontal distance from $A$`A` to $B$`B`:

$\text{run}=7-3=4$run=7−3=4.

Notice that we subtracted the $x$`x`-values and the $y$`y`-values in the same order - we check our sketch, and it does seem sensible that between $A$`A` and $B$`B` there is a rise of $-8$−8 and a run of $4$4. We can now put these values into our formula to find the gradient:

$\text{gradient }$gradient | $=$= | $\frac{\text{rise }}{\text{run }}$rise run |

$=$= | $\frac{-8}{4}$−84 | |

$=$= | $-2$−2 |

We have a negative gradient, as we suspected. Now we know that when we travel along this line a step of $1$1 in the $x$`x`-direction means a step of $2$2 **down** in the $y$`y`-direction.

Horizontal lines have no rise value. The $\text{rise }=0$rise =0.

So the gradient of a **horizontal **line is $\text{Gradient }=\frac{\text{rise }}{\text{run }}$Gradient =rise run $=$=$\frac{0}{\text{run}}$0run$=$=$0$0.

Vertical lines have no run value. The $\text{run }=0$run =0.

So the gradient of a **vertical **line is $\text{gradient }=\frac{\text{rise }}{\text{run }}$gradient =rise run $=$=$\frac{\text{rise }}{0}$rise 0. Division by $0$0 results in the value being **undefined**.

Gradient formula

For any line, $\text{gradient }=\frac{\text{rise }}{\text{run }}$gradient =rise run

To calculate the rise from two points, take the difference of the $y$`y`-values (subtract left from right)

To calculate the run from two points, take the difference of the $x$`x`-values (subtract left from right)

The gradient of a horizontal line is $0$0. The gradient of vertical line is **undefined**.

What is the gradient of the line shown in the graph, given that Point A $\left(3,3\right)$(3,3) and Point B $\left(6,5\right)$(6,5) both lie on the line?

What is the gradient of the line going through A $\left(-1,1\right)$(−1,1) and B $\left(5,2\right)$(5,2)?

The gradient of interval AB is $3$3. A is the point ($-2$−2, $4$4), and B lies on $x=3$`x`=3. What is the $y$`y`-coordinate of point B, denoted by $k$`k`?