Linear Equations

Lesson

In Keeping it in Proportion, we learnt about direct proportion, where one amount increased at the same constant rate as the other amount increased. Now we are going to look an inverse proportion.

Inverse proportion means that as one amount *increases* the other amount *decreases*. Mathematically, we write this as $y\propto\frac{1}{x}$`y`∝1`x`. For example, speed and travel time are inversely proportional because the faster you go, the shorter your travel time.

General Equation for Amounts that are Inversely Proportional

We express these kinds of inversely proportional relationships generally in the form

$y=\frac{k}{x}$`y`=`k``x`

where $k$`k` is the constant of proportionality and $x$`x` and $y$`y` are any variables

Just like identifying directly proportional relationships, when identifying inversely proportional relationships we need to find the constant of proportionality or the constant term that reflects the rate of change. The we can substitute any quantity we like into our equation.

Let's see how in the examples below.

Consider the equation $s=\frac{375}{t}$`s`=375`t`.

State the constant of proportionality.

Find the value of $s$

`s`when $t=6$`t`=6. Give your answer as an exact value.Find the value of $s$

`s`when $t=12$`t`=12. Give your answer as an exact value.

Consider the values in each table. Which of them could represent an inversely proportional relationship between $x$`x` and $y$`y`?

$x$ `x`$1$1 $2$2 $3$3 $4$4 $y$ `y`$3$3 $1.5$1.5 $1$1 $0.75$0.75 A$x$ `x`$1$1 $2$2 $3$3 $4$4 $y$ `y`$36$36 $18$18 $12$12 $9$9 B$x$ `x`$1$1 $5$5 $6$6 $10$10 $y$ `y`$3$3 $75$75 $108$108 $300$300 C$x$ `x`$1$1 $2$2 $3$3 $4$4 $y$ `y`$4$4 $5$5 $6$6 $7$7 D$x$ `x`$1$1 $2$2 $3$3 $4$4 $y$ `y`$3$3 $1.5$1.5 $1$1 $0.75$0.75 A$x$ `x`$1$1 $2$2 $3$3 $4$4 $y$ `y`$36$36 $18$18 $12$12 $9$9 B$x$ `x`$1$1 $5$5 $6$6 $10$10 $y$ `y`$3$3 $75$75 $108$108 $300$300 C$x$ `x`$1$1 $2$2 $3$3 $4$4 $y$ `y`$4$4 $5$5 $6$6 $7$7 D