Problem 27 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=5.5, y=1.1$$

Step-by-Step Solution

Answer

The equation that relates x and y in this direct variation problem is \(y = 0.2x\).

Step 1: Identify the Known Variables

The problem provides the values of x and y. In this case, \(x = 5.5\) and \(y = 1.1\).

Step 2: Substitute the Known Values into the Direct Variation Equation

For direct variation, \(y=kx\). Let's substitute the provided values into this equation. This gives \(1.1=k*(5.5)\).

Step 3: Solve for the Constant of Variation

To find the constant of variation, we solve for \(k\) in the equation \(1.1=k*(5.5)\). Dividing both sides of the equation by 5.5, we find that \(k = 1.1 / 5.5 = 0.2\).

Step 4: Write the Direct Variation Equation

Now that the constant of variation, \(k\), has been found, the direct variation equation can be written as \(y = 0.2x\).

Key Concepts

Constant of Variation

In direct variation, the constant of variation, denoted by "k," plays a crucial role. It represents the rate at which one variable changes in relation to another. When two variables, say \(x\) and \(y\), vary directly, their relationship can be expressed as \(y = kx\). This means that \(y\) is directly proportional to \(x\), with \(k\) being the multiplier that scales \(x\) to equal \(y\).
To find the constant of variation, you need specific pairs of \(x\) and \(y\) values. By plugging these values into the formula \(y = kx\), you calculate \(k\) by solving the equation. Simply divide \(y\) by \(x\) to isolate \(k\):

• If the variables \(x = 5.5\) and \(y = 1.1\) are given, substitute these numbers into the equation: \(1.1 = k \times 5.5\).
• Solving for \(k\) yields \(k = \frac{1.1}{5.5} = 0.2\).

This constant tells you how much \(y\) changes for a unit change in \(x\). It's an essential piece that defines the direct variation between the two variables.

Equation Writing

Once you have identified the constant of variation, the next step is to write the equation that accurately shows the relationship between the variables \(x\) and \(y\). This equation is important because it provides a simple yet powerful way to predict and understand how the variables interact.
In our example, after finding \(k = 0.2\), the equation becomes \(y = 0.2x\). This equation is the direct variation formula specific to the given values of \(x\) and \(y\). It indicates that for every unit increase in \(x\), \(y\) increases by 0.2 units.
This step involves a straightforward assembly of the direct variation equation using the constant. It establishes how the two variables are related through the lens of proportional change. Therefore, the equation \(y = 0.2x\) succinctly embodies this linear relationship by representing \(y\) as a direct function of \(x\).

• Remember, writing an equation is about clearly establishing the relationship between variables using mathematical expressions.
• In direct variation, always ensure your equation follows the form \(y = kx\), where \(k\) is your calculated constant.

Problem-Solving Steps

Effective problem-solving in mathematics often requires a structured approach. Let's explore the fundamental steps you can follow when solving direct variation problems, like the one in our example. Following these steps helps you tackle problems systematically and with confidence.

Identify Given Information

The first step in any problem-solving process is to gather and note down all the given information. For direct variation, this means identifying the values of \(x\) and \(y\). This helps set the stage for subsequent steps.

Formulate the Direct Variation Equation

Given the equation \(y = kx\), substitute the known values into this formula. This will typically result in an equation where you need to solve for \(k\). Take the given values \(x = 5.5\) and \(y = 1.1\) and substitute them to form \(1.1 = k \times 5.5\).

Solve for the Constant

With your equation formed, solve for the constant of variation. This involves algebraic manipulation, like dividing both sides by \(x\), to isolate \(k\). This step yields \(k = \frac{1.1}{5.5} = 0.2\).

Write the Final Equation

Once you have \(k\), you can write the direct variation equation as \(y = 0.2x\). This final equation is a succinct representation of the relationship between the variables.

• By following these steps, you ensure a clear and logical approach to solving direct variation problems.
• Understanding each phase helps you build confidence in tackling similar questions in the future.