Problem 55 - Algebra 1

Question

The variables \(x\) and \(y\) vary directly. Use the given values of the variables to write an equation that relates \(x\) and \(y .\) $$x=-2, y=-1$$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) with direct variation is \(x = 2y\).

Step 1: Understand the problem and the given

Direct variation means that the ratio between the two variables is constant. The relationship can be written as \(x = k \cdot y\), where \(k\) is the constant of variation. Given \(x = -2\) and \(y = -1\), we can substitute these values into the equation to solve for the constant \(k\) and thus find the relationship between \(x\) and \(y\).

Step 2: Substitute the given values

Substitute \(x = -2\) and \(y = -1\) into the equation \(x = k \cdot y\) to solve for \(k\). This gives \(-2 = k \cdot (-1)\). Solving this equation for \(k\) gives \(k = 2\).

Step 3: Write the equation

Now, replace \(k\) in the direct variation equation with the value found in step 2. The equation that relates \(x\) and \(y\) is therefore \(x = 2y\).

Key Concepts

Constant of Variation

The concept of constant of variation is a foundational idea in direct variation relationships. It refers to the unchanging number that relates two variables that vary directly. Think of it as the multiplier that connects one variable to another in a consistent manner. In the given problem, we find this constant by dividing the value of one variable by the corresponding value of the other variable. When given that \(x = -2\) and \(y = -1\), we deduce that the constant of variation \(k\) is \(2\), because \(-2 = k \times -1\). Understanding the role of the constant can greatly simplify the approach to solving direct variation problems.

To ensure a thorough grasp of this concept, remember that this constant will always be the same for a pair of variables in a direct variation, regardless of their individual values. By identifying this constant, we can predict the value of one variable given the other, which is a powerful tool in algebra.

Algebraic Equations

Algebraic equations are the bread and butter of solving mathematical problems involving relationships between variables. They are statements of equality that include variables, coefficients, and often constants. An equation represents a mathematical sentence declaring that two expressions are equal. Particularly in the context of direct variation, the algebraic equation is often simple and linear, setting the product of one variable and the constant of variation equal to the other variable—expressed as \(x = k \times y\).

In the exercise, the goal is to create such an equation that represents the relationship between \(x\) and \(y\). By substituting the given values into the direct variation equation and solving for the constant of variation, we craft an algebraic equation that models the relationship accurately. Learning how to construct and manipulate these equations is essential for success in algebra and beyond.

Proportional Relationships

Proportional relationships are all about the idea of two quantities increasing or decreasing in lockstep. They describe how one quantity changes at the same rate as another. This is precisely what happens in direct variation, where one variable is a constant multiple of the other. When we speak of two quantities being proportional, we mean that they are related in such a way that the ratio of the two quantities remains constant.

For instance, in our example where \(x = 2y\), doubling \(y\) will always double \(x\), and halving \(y\) will in turn halve \(x\). Recognizing proportional relationships allows students to make quick predictions and understand the scale of changes in quantities. It's a concept deeply rooted not only in algebra but also in practical scenarios such as calculating ingredient amounts in recipes or converting currencies during travel.