To determine in which table y varies directly with x, compare the ratio (y \div x) for every row. If the ratio is the same for all nonzero (x)-values, then (y) varies directly with (x), and the relationship can be written as (y = kx), where (k) is the constant of variation And that's really what it comes down to..
Understanding Direct Variation
A relationship shows direct variation when one quantity changes at a constant rate in relation to another. In algebra, this relationship is commonly written as:
[ y = kx ]
In this equation:
- (x) is the independent variable.
- (y) is the dependent variable.
- (k) is the constant of variation.
- (k) must be a nonzero constant.
When (y) varies directly with (x), increasing (x) causes (y) to increase by the same proportional amount. To give you an idea, if (x) doubles, (y) also doubles. If (x) triples, (y) triples. This proportional behavior is the key feature of direct variation.
A direct variation relationship always has these characteristics:
- The equation can be written in the form (y = kx).
- The ratio (\frac{y}{x}) is constant.
- The graph is a straight line passing through the origin ((0,0)).
- There is no added constant, so the equation is not (y = kx + b) unless (b =
