Question

Tell whether \(x\) and \(y\) are in a proportional relationship. \(y=\frac{x}{2}\)

Short answer

Yes, \(x\) and \(y\) are in a proportional relationship because the given equation could be written in the form \(y=kx\), where \(k\) is a constant.

Step-by-step solution

Identify the form of the given equation

The given equation \(y=\frac{x}{2}\) is in the form \(y=kx\), where \(k\) is the coefficient of \(x\). Here, \(k=\frac{1}{2}\).

Draw the conclusion

Since the given equation could be written in the form \(y=kx\), where \(k\) is constant, we can conclude \(x\) and \(y\) are in a proportional relationship.

Key concepts

Proportional Relationship in Algebra

Learning about proportional relationships in algebra enables students to connect mathematical concepts with real-world scenarios. In essence, a proportional relationship between two variables means that they change at the same rate. When one quantity increases or decreases, the other does so in a directly proportional manner.

For instance, consider the equation from your exercise: \(y = \frac{x}{2}\). The ratio of y to x is constant, which is fundamental in defining a proportional relationship. If you imagine this as a recipe, if you doubled the amount of ingredient x, the amount of outcome y would also double. This characteristic is crucial in determining proportionality and can be visualized graphically as a straight line passing through the origin with a slope that represents the constant rate of change. Understanding the connection between the algebraic representation and its graphical implications is a vital skill in interpreting and solving proportional problems in algebra.

Constant of Proportionality

The constant of proportionality is the factor that relates two variables that are in a proportional relationship. It is represented by the variable \(k\) in the general form \(y=kx\). The value of \(k\) indicates how much y changes for a one unit change in x.

In the exercise, the equation \(y = \frac{x}{2}\) presents us with a constant of proportionality of \(\frac{1}{2}\). This means that for every 1 unit increase in x, y increases by half of its value. Recognizing \(k\) and understanding how it functions is like knowing the 'price per item' when shopping; it tells us the rate at which two quantities are connected.

It is also important to remember that the constant of proportionality in a proportional relationship will always be a constant value, signifying a linear and predictable pattern of change between the variables.

Proportional Equations

In algebra, proportional equations serve as the foundation to establish the relationship between proportional variables. They take the form \(y=kx\), where \(k\) remains consistent regardless of the value of x and y. This consistent ratio is what establishes the equation's proportionality.

When you're given a proportional equation, like in your exercise where \(y=\frac{x}{2}\), your goal is to verify that y and x change at a constant rate, which is \(k\). If this rate stays consistent for all values of x and y, then you have verified a proportional relationship. In real-world applications, proportional equations allow us to calculate one quantity based on another—like converting miles to kilometers or using a recipe to bake a larger cake. Grasping how these equations work provides a powerful tool for solving a wide array of practical problems.