Rate of change.

Step 1 of 4

The candle

A candle $12$⁠ cm tall is burning, and every hour it gets $1.5$⁠ cm shorter. Its height $y$⁠ after $x$⁠ hours is

$y = 12 - 1.5x$⁠

That is a line: $n = 12$⁠ is the height at the start, and $m = -1.5$⁠ is the rate of change: how much the height changes every hour. The slope is negative because the candle gets shorter.

Step 2 of 4

When does it run out?

The candle runs out when its height is zero, that is, where the line meets the $x$⁠-axis. That is the equation $12 - 1.5x = 0$⁠.

Step 3 of 4

Two candles

Next to it burns a second candle, shorter but slower: it starts at $9$⁠ cm and gets $0.75$⁠ cm shorter every hour, $y = 9 - 0.75x$⁠. It is the pink line. The first candle starts taller but goes down faster, so at some point they will be the same height.

Step 4 of 4

Units of slope

A slope has units: the units of $y$⁠ per unit of $x$⁠. Here $y$⁠ is in centimetres and $x$⁠ in hours, so the slope is $-1.5$⁠ centimetres per hour. Change Change per hour and Starting height, and see when a candle like that runs out.