Question

$y$ is directly proportional to $x$ . Write an equation that shows the relationship if $x=2$ and $y=6$ ?

Answer 1

Hint: To solve such problems always remember that directly proportional means $y=Kx$ where $K$ is any constant. Here the equation can be formed by substituting the values of $x$ and $y$ in $y=Kx$ . This will give the value of the constant and thus the equation.

Complete step-by-step solution:
Given that $y$ is directly proportional with $x$ . That is,
$y=Kx$ where $K$ is any constant.
Also given that $x=2$ and $y=6$ .
Substitute this in $y=Kx$ . That is,
  $6=K\times 2$
$K=3$
Therefore, the value of the constant is $K=3$ .

Hence the equation that shows the relationship if $x=2$ and $y=6$ is as shown below
$y=3x$.

Additional Information: Suppose that there are two quantities $x$ and $y$ . If these two quantities increase or decrease at a constant rate then they are said to be directly proportional. If $x$ and $y$ are directly proportional then it can be written as $x\alpha y$ . Here $\alpha$ means proportional to.
If an increase in $x$ causes a decrease in $y$ or vice versa then the quantities are said to be in inverse proportion. If $x$ and $y$ are inversely proportional then it can be written as $x\alpha {\displaystyle \frac{1}{y}}$ . If $y$ is inversely proportional to $x$ , then it can be written as $y={\displaystyle \frac{K}{x}}$ , where $K$ is any constant. Here $K$ is called the constant of proportionality of the relationship.

Note: Always remember that if the question says $y$ is directly proportional to $x$ , then it means $y$ will be a constant multiple of $x$ . It can also be said that if $y$ is directly proportional to $x$ , then ${\displaystyle \frac{y}{x}}=K$ or $y=Kx$ . If $x_1,x_2$ are the values corresponding to $x$ and $y_1,y_2$ are the values corresponding to $y$, also if $y$ is directly proportional to $x$ , then ${\displaystyle \frac{y_1}{x_1}}={\displaystyle \frac{y_2}{x_2}}$ .