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Proportional Relationships

A proportional relationship is one in which two quantities vary directly with each other. We say the variable $y$ varies directly as $x$ if:

$y = k x$

for some constant $k$ , called the constant of proportionality .

(Some textbooks describe a proportional relationship by saying that " $y$ varies proportionally with $x$ " or that " $y$ is directly proportional to $x$ .")

This means that as $x$ increases, $y$ increases and as $x$ decreases, $y$ decreases-and that the ratio between them always stays the same.

The graph of the proportional relationship equation is a straight line through the origin.

Example 1:

Given that $y$ varies proportionally with $x$ , with a constant of proportionality $k = \frac{1}{3}$ , find $y$ when $x = 12$ .

Write the equation of the proportional relationship.

The variable $x$ varies proportionally with $y$ with a constant of proportionality equal to $\frac{1}{3}$ .

So, Math diagram

Substitute the given $x$ value.

Math diagram

Example 2:

Given that $y$ varies proportionally with $x$ , find the constant of proportionality if $y = 24$ and $x = 3$ .

Write the equation of the proportional relationship.

$y = k x$

Substitute the given $x$ and $y$ values, and solve for $k$ .

$\begin{array}{l} 24 = k \cdot 3 \\ k = 8 \end{array}$

Example 3:

Suppose $y$ varies proportionally with $x$ , and $y = 30$ when $x = 6$ . What is the value of $y$ when $x = 100$ ?

Write the equation of the proportional relationship.

$y = k x$

Substitute the given $x$ and $y$ values, and solve for $k$ .

$30 = k \cdot 6$

$k = 5$

The equation is $y = 5 x$ . Now substitute $x = 100$ and find $y$ .

$\begin{array}{l} y = 5 \cdot 100 \\ y = 500 \end{array}$