Here we look at a special kind of linear relationship called direct variation.
Variation deals with the way two or more variables interact with each other, and describe how a change in one variable results in a change in the other variable. There are different types of variation, with the main ones being direct variation and inverse variation. Here we consider direct variation.
Direct variation is when a change in one variable leads to a directly proportional change in the other variable.
Let's say we have two variables, $x$x and $y$y. If $x$x is directly proportional to $y$y, then an increase in $x$x, will lead to a proportional increase in $y$y. In a similar way, a decrease in $x$x, will lead to a proportional decrease in $y$y. This direct variation relationship can be written as:
$y\propto x$y∝x
where the symbol $\propto$∝ means 'is directly proportional to'.
As another example, the statement:
"Earnings, $E$E, are directly proportional to the number of hours, $H$H, worked."
could be written as:
$E\propto H$E∝H
For the purposes of calculation, we can turn a proportionality statement into an equation using a constant of variation (or constant or proportionality).
To solve a direct variation problem, the key step is to find the constant of variation, $k$k.
The cost of repairing a bicycle is directly proportional to the amount of time spent working on it. If it takes $3$3 hours to complete a repair job that costs $\$255$$255, calculate
Solution
We will use $C$C for the cost of repairs and $T$T for the time taken.
$C$C | $\propto$∝ | $T$T |
$C$C | $=$= | $kT$kT |
$C$C | $=$= | $kT$kT |
|
$255$255 | $=$= | $k\times3$k×3 |
(Substitute $C=255$C=255 and $T=3$T=3) |
$255$255 | $=$= | $3k$3k |
|
$\frac{255}{3}$2553 | $=$= | $\frac{3k}{3}$3k3 |
(Divide both sides by $3$3) |
$85$85 | $=$= | $k$k |
|
$k$k | $=$= | $85$85 |
|
$C$C | $=$= | $85T$85T |
|
$C$C | $=$= | $85\times2.5$85×2.5 |
(Substituting $T=2.5$T=2.5) |
$C$C | $=$= | $\$212.50$$212.50 |
$C$C | $=$= | $85T$85T |
|
$357$357 | $=$= | $85T$85T |
(Substituting $C=357$C=357) |
$\frac{357}{85}$35785 | $=$= | $\frac{85T}{85}$85T85 |
|
$4.2$4.2 | $=$= | $T$T |
|
$T$T | $=$= | $4.2$4.2 hours |
|
Consider the equation $P=90t$P=90t.
State the constant of proportionality.
Find the value of $P$P when $t=2$t=2.
Find the equation relating $a$a and $b$b given the table of values.
$a$a | $0$0 | $1$1 | $2$2 | $3$3 |
---|---|---|---|---|
$b$b | $0$0 | $2$2 | $4$4 | $6$6 |
If $y$y varies directly with $x$x, and $y=\frac{1}{5}$y=15 when $x=4$x=4:
Using the equation $y=kx$y=kx, find the variation constant, $k$k.
Enter each line of work as an equation.
Hence, find the equation of variation of $y$y in terms of $x$x.
The number of revolutions a wheel makes varies directly with the distance it rolls. A bike wheel revolves $r$r times in $t$t seconds.
If the wheel completes $40$40 revolutions in $8$8 seconds, find the value of $k$k.
Hence express $r$r in terms of $t$t.
How many revolutions will the wheel complete in $7$7 seconds?
Some direct variation problems can be solved without using equations.
$3.1$3.1kg of pears cost $\$7.13$$7.13. How much would $1.5$1.5kg cost?
A graph representing direct variation, will always be a straight line that passes through the origin $\left(0,0\right)$(0,0). In other words, its vertical intercept will always be zero. The gradient of the line will be equal to the constant of variation.
The diagram below show a linear graph, where variable $2$2 is directly proportional to variable $1$1.
Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol.
Number of litres ($x$x) | $0$0 | $10$10 | $20$20 | $30$30 | $40$40 | $50$50 |
---|---|---|---|---|---|---|
Cost of petrol ($y$y) | $0$0 | $16$16 | $32$32 | $48$48 | $64$64 | $80$80 |
How much does petrol cost per litre?
Write an equation linking the number of litres of petrol pumped ($x$x) and the cost of the petrol ($y$y).
How much would $65$65 litres of petrol cost at this unit price?
Graph the equation $y=1.6x$y=1.6x.
In the equation, $y=1.6x$y=1.6x, what does $1.6$1.6 represent?
The total cost of petrol pumped.
The number of litres of petrol pumped.
The unit rate of cost of petrol per litre.