Proportional Relationships

Recall that a proportional relationship can be represented by an equation of the form *y* = k*x*, where k is a constant.

The constant, k, is referred to as the constant of proportionality, the unit rate, and/or the **slope**.

Notice in the equation *y* = k*x*, if *x* equals 1, then *y* equals k.

So, on the graph of a proportional relationship, when *x* equals 1, the corresponding *y*-value is the slope.

In the equation *y* = k*x*, if *x* equals 0, then *y* also equals 0.

So, on a coordinate grid, a line drawn through points which represent a proportional relationship, and extended through the *x*-axis, contains the point (0, 0).

Also notice, in the equation *y* = k*x*, if we divide both sides by *x*, we get ^{y}/_{x} = k.

So, for a given (*x*, *y*) in a proportional relationship, the slope is the ratio of *y* to *x*, *x* ≠ 0.

**EXAMPLE**
A pound of fudge costs three different prices at three different candy stores. The representations below show the cost, *y*, based on the number of pounds of candy, *x*, at the three stores. For each representation, identify and interpret the slope.STORE A | | STORE B | | STORE C |

| |
Pounds (*x*) |
Cost (*y*) |
1 | 11 |
2 | 22 |
3 | 33 |
4 | 44 |
| | |

For an equation of the form *y* = k*x*, k is the slope.

For the given equation, the slope is 16.

In this situation, the slope can be interpreted as $16 per pound of candy.

In the table, the ratio of each *y* to its corresponding *x* is 11.

So, the slope of the line represented by the table is 11.

In this situation, the slope can be interpreted as $11 per pound of candy.

The graph represents a proportional relationship.

On the graph of a proportional relationship, when *x* equals 1, the corresponding *y*-value is the slope. On this graph, when *x* equals 1, *y* equals 8. So, the slope is 8.

The slope can be interpreted as $8 per pound of candy.