Question

Explain what it means for x and y to vary directly.

Short answer

If x and y vary directly, it means that the relationship between x and y is proportional. When one variable increases, the other increases as well, and when one decreases, the other also decreases. This relationship can be mathematically represented as \(y = kx\), where \(k\) is the constant of variation.

Step-by-step solution

Define Direct Variation

Direct variation is a relationship between two variables in which one is a constant multiple of the other. In other words, when one variable changes the other one does as well, in the same direction. One way to put this is to say that variable \(y\) varies directly as \(x\), if \(y = kx\) for some constant \(k\), known as the constant of variation.

Discuss Proportional Relationship

For two quantities to be in direct variation, a proportional relationship must exist. This is to say if \(y\) varies directly as \(x\), there is a number \(k\) such that the ratio \(\(y/x\)\) is always the same. This constant number \(k\) is the constant of proportionality. If the value of one variable is multiplied by any amount, the other must be multiplied by the same amount in order for the equation to stay balanced.

Provide Examples

For instance, if we say that the distance (d) travelled varies directly as time (t), it means that travelling for 2 hours will cover twice the distance of travelling for 1 hour, assuming constant speed. Similarly, if the number of people (n) in a group varies directly with the total cost (c), it means that doubling the number of people will double the total cost, assuming the cost per person remains the same. In both instances, one variable changes proportionally with the other, which is the key part in direct variation.

Key concepts

Constant of Variation

Imagine you are trying to figure out how many apples you can buy with a certain amount of money. The amount of apples (\(y\)) you get directly depends on how much money (\(x\) ) you have, given apples cost the same. The constant of variation is like the price tag on those apples; it is the unchanging number that links your money to the number of apples you can buy. Mathematically, in the relationship \(y = kx\), the constant (\(k\) ) is the constant of variation. It tells us how much \(y\) will change for every one unit change in \(x\). If \(k\) is 3, for every dollar you spend, you get 3 apples. This number is crucial as it tells us the rate at which variables are connected and signifies their direct dependency on each other.

Proportional Relationship

Think about baking cookies. You know that the ratio of ingredients you use directly impacts how your cookies will taste. In math, we talk about a proportional relationship in a similar vein. It's an equation that shows how two variables increase or decrease consistently with each other. For instance, if you're told that the number of cookies (\(y\) ) you can bake is directly proportional to the cups of flour (\(x\) ) you have, then as you double your flour, you're also doubling your cookie output. In direct variation, represented by \(y = kx\), it means that for any amount of flour, you simply multiply it by the constant of proportionality (\(k\) ) to determine the number of cookies. If \(k\) is 10, then 2 cups of flour give you 20 cookies, and 3 cups give you 30 cookies, and so on. This predictable pattern where the quotient \(y/x\) remains the same is what defines a proportional relationship.

Constant of Proportionality

Now, let's tie back to our cookie baking example. The constant of proportionality is like the specific recipe that dictates the exact amount of cookies we get per cup of flour. It's the fixed number that we multiply the quantity of one variable by, to get the quantity of the other in a direct variation scenario. So, if we represent the number of cookies (\(y\) ) as a multiple of cups of flour (\(x\) ), with the equation \(y = kx\), the constant (\(k\) ) you're using is the constant of proportionality. It's a special kind of 'recipe' number because it doesn't change, no matter how many cups of flour you start with. If your 'recipe' says 5 cookies per cup of flour (\(k = 5\) ), then 100 cups of flour will yield 500 cookies, maintaining that same ratio. The beauty of the constant of proportionality is that it keeps everything balanced, predictable, and harmonious in math, just like a well-followed recipe does in baking.