Identifying Proportional Relationships

A proportional relationship is a relationship between two quantities, \(x\) and \(y\), where one is always a constant multiple of the other. In equation form, this looks like \(y = kx\), where \(k\) is called the constant of proportionality. As \(x\) increases, \(y\) increases (or decreases) by the same fixed factor every time.

Because \(y = kx\) is a specific type of linear equation, every proportional relationship is linear, but not every linear relationship is proportional. If you want a refresher on what makes an equation linear in the first place, see the lesson on the introduction to linear equations.

You can identify a proportional relationship from any of these three representations:

• Equation: it must simplify to \(y = kx\), with no number added or subtracted.
• Table of values: the ratio \(\frac{y}{x}\) must be the same constant for every pair.
• Graph: it must be a straight line that passes through the origin \((0,0)\).

To check an equation, get \(y\) alone on one side and look at what remains on the other side.

Example: \(y = 5x\) is proportional. Here \(k = 5\), and there is no extra constant term.

Example: \(y = 5x + 2\) is not proportional. The "+2" means \(y\) is never simply a multiple of \(x\); it also shifts the line up, so it can't pass through the origin.

If an equation is given in a different form, such as \(2x + y = 0\) or a two-step form like \(3y - 6x = 0\), rearrange it to solve for \(y\) first. This is the same skill used in solving two step linear equations, and it makes the constant of proportionality easy to spot once \(y\) is isolated.

In a proportional table, dividing \(y\) by \(x\) gives the same number every single time.

\(x\)	\(y\)	\(y \div x\)
2	6	3
4	12	3
6	18	3

Since \(y \div x = 3\) in every row, this table represents the proportional relationship \(y = 3x\). If even one ratio is different from the rest, the relationship is not proportional. For more practice reading and building tables like this one, check out the lesson on the table of values.

Compare that to a non-proportional table:

\(x\)	\(y\)	\(y \div x\)
2	7	3.5
4	11	2.75
6	15	2.5

The ratio changes every row, so this table does not describe a proportional relationship (its rule is actually \(y = 2x + 3\)).

A graph shows a proportional relationship when it is a straight line that passes directly through the origin, \((0,0)\). If the line is shifted up or down so it crosses the \(y\)-axis anywhere else, it is not proportional, even if it is still a straight line.

Notice that both graphs are straight lines with the same slope, but only the first one is proportional, because only that one passes through \((0,0)\).

A recipe uses 3 cups of flour for every 2 cups of sugar. Is the relationship between flour \((x)\) and sugar \((y)\) proportional?

Check the ratio: \(y \div x = \frac{2}{3}\). Testing with double the amounts, 6 cups of flour and 4 cups of sugar, gives \(4 \div 6 = \frac{2}{3}\) as well. The ratio stays constant, so yes, this is proportional, with equation \(y = \frac{2}{3}x\).

Watch out for these situations, which often get confused with proportional relationships:

• A linear equation with a constant added, like \(y = x + 4\), looks similar to \(y = kx\) but is not proportional.
• A table where the \(y\)-values increase by the same amount each time (not the same ratio) is usually not proportional.
• A graph that is a straight line but does not touch the origin is never proportional, no matter how close it comes.

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