Algebra

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Direct Proportion

Values that change relative to each other in a regular way can be described as proportional. Algebraic values can also be proportional, and we can use the constant of proportionality to find unknown values.

If two quantities are directly proportional, the rate at which one increases is proportional to the other. This means that the ratio between the two quantities is constant.

If two quantities are proportional, this means that...

If $y$ is proportional to $x$ :

$y = kx$

where $k$ will be a number. $k$ is the constant of proportionality, and represents the ratio between the two proportional values.

For example, if $y$ is proportional to $x$ and $y = 10$ when $x = 5$ , what is the value of $k$ ?

$y=kx$

Rearrange the equation to make $k$ the subject

Since $y=kx$ , we can divide both sides by $x$ to find that $k=\dfrac{y}{x}$

Put the values we know into the equation

$\dfrac{10}{5} = 2$ . Therefore, we know that $k=2$ .

Form an equation

We can form an equation with our value of $k=2$ : $y = 2x$ . We can use this to find other values of $y$ .

If $y=2x$ , what is $y$ when $x = 20$ ?

When $x=20$ , $y=40$

We can find this by putting $x=20$ into our equation: $y=2\times 20$ .

$y=kx$

$y = 4$ when $x = 8$ . Find $k$

$y=0.5x$

If $y = 12$ , what is $x$ ?

Now we know how to find $k$ , we can use it to find the value of an unknown $x$ or $y$ in algebraic problems.

$y$ is proportional to $x^3$ . When $x = 2$ , $y = 40$ . What is $y$ when $x = 3$ ?

The two quantities are proportional

Therefore, we can express them as $y=kx^3$ .The first step is to find the value of $k$ , so we can apply this to find $y$ when $x=3$ .

To find $k$ , put the values in the equation

We are given the values $x=2$ and $y=40$ . Therefore, using our equation, $40=k(2^3)$ , so $40=k(8)$ .

Rearrange to make $k$ the subject

To isolate $k$ , we need to divide both sides by 8. $40=k(8)$ , so $k=\dfrac{40}{8}$ . Therefore, $k=5$ .

Use $k$ to find $y$ when $x=3$

Now we know $k$ , we can substitute this into our equation when $x=3$ . Therefore, the equation is $y=5 \times 3^3$ , so $y=135$ .