Problem 5 - Algebra A Combined Function

Question

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=7\) when \(x=\frac{1}{2}\)

Step-by-Step Solution

Answer

The constant of variation is 14, and the equation is \(y = 14x\).

Step 1: Understanding Direct Variation

In direct variation, if we know that \(y\) varies directly as \(x\), it means \(y = kx\) for some constant \(k\), called the constant of variation. Here, we need to find \(k\) when \(y = 7\) and \(x = \frac{1}{2}\).

Step 2: Substitute Known Values

Substitute the given values of \(y=7\) and \(x=\frac{1}{2}\) into the direct variation formula \(y = kx\).The equation becomes:\(7 = k \left( \frac{1}{2} \right)\).

Step 3: Solve for the Constant of Variation

To find \(k\), divide both sides of the equation \(7 = k \left( \frac{1}{2} \right)\) by \(\frac{1}{2}\):\[k = \frac{7}{\frac{1}{2}} = 7 \cdot 2 = 14\]

Step 4: Write the Direct Variation Equation

Now that we have the constant of variation \(k = 14\), write the direct variation equation as\(y = 14x\).

Key Concepts

Constant of Variation

In the world of mathematics, especially when dealing with direct variation, the concept of "constant of variation" is pivotal. This is the number that remains consistent in the equation when two variables are directly proportional to one another. In simpler terms, when we say that the value of \(y\) changes with the value of \(x\), there is a constant rate, or factor, determining how \(y\) scales with \(x\).
To find this constant, you need to have a scenario where you already know the values of \(y\) and \(x\). Using the formula for direct variation, \(y = kx\), you substitute the known values into the equation. In our exercise, with \(y = 7\) and \(x = \frac{1}{2}\), we determine \(k\) by solving \(7 = k \left( \frac{1}{2} \right)\). Solving for \(k\) gives us 14, so 14 is the constant of variation. This number 14 tells us that for every increase by one unit in \(x\), \(y\) increases by 14 units.

Direct Variation Equation

Once the constant of variation is established, we can easily construct the direct variation equation. This equation is a straightforward representation that illustrates the relationship between two variables: \(y\) and \(x\). The standard form of this equation is \(y = kx\), where \(k\) is the constant of variation.
Given the previously calculated \(k\) value of 14, your direct variation equation becomes \(y = 14x\). This equation lets you find \(y\) for any given value of \(x\) and vice versa.
For instance:

• If \(x = 1\), then \(y = 14 \times 1 = 14\)
• If \(x = 2\), then \(y = 14 \times 2 = 28\)

This way, the direct variation equation is very handy for making predictions and understanding the linear relationship between the two variables.

Proportional Relationships

Proportional relationships are all around us, and understanding this concept can open up new ways to comprehend everyday occurrences. In mathematics, a proportional relationship means that two quantities scale with one another at a constant rate.
This constant rate is, once again, our "constant of variation." When we say that \(y\) is directly proportional to \(x\), the ratio \(\frac{y}{x}\) will always equal the constant of variation, \(k\). This implies that whenever \(x\) is multiplied or divided by a number, \(y\) will change by the same multiplication or division but scaled by \(k\).In summary, saying that \(y\) changes proportionally with \(x\) implies:

• Both variables increase together at a constant rate.
• The graph, if plotted, will always pass through the origin (0,0).
• This type of relationship is predictable and useful in various real-world applications.

Grasping proportional relationships and how they manifest in direct variation gives us a powerful tool for reasoning and solving practical problems.