Question

a candle burns at a steady rate per hour. the relationship between y, the height of the candle and x, the number of minutes it has been burning is shown in the table. write an equation to represent the relationship between x and y.

time (minutes)	height of candle (in)
2	7.6
3	7.4
5	7

look at the y - values. what is happening to the y - values as x increases? if the y - values are decreasing, what does that tell us about the rate?
practice: complete the table to satisfy the equation y = 0.5x - 5.

x	y
2
4	- 3
12
15

Explanation:

Step1: Find the slope for the candle - burning problem

As \(x\) (time in minutes) increases by 1, \(y\) (height of candle) decreases by \(7.8 - 7.6=0.2\). So the slope \(m=- 0.2\). When \(x = 1\), \(y = 7.8\). Using the point - slope form \(y - y_1=m(x - x_1)\), with \(x_1 = 1\) and \(y_1 = 7.8\), we have \(y-7.8=-0.2(x - 1)\), which simplifies to \(y=-0.2x + 8\).

Step2: Complete the table for \(y = 0.5x-5\)

When \(x = 0\), substitute \(x\) into the equation: \(y=0.5\times0 - 5=-5\).
When \(x = 2\), \(y=0.5\times2-5=1 - 5=-4\).
When \(x = 12\), \(y=0.5\times12-5=6 - 5 = 1\).
When \(x = 15\), \(y=0.5\times15-5=7.5 - 5 = 2.5\).

Answer:

For the candle - burning problem, the equation is \(y=-0.2x + 8\).
For the table completion:

\(x\)	\(y\)
\(2\)	\(-4\)
\(4\)	\(-3\)
\(12\)	\(1\)
\(15\)	\(2.5\)