Problem 55 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=4.6, y=1.2$$

Step-by-Step Solution

Answer

The direct variation equation that relates x and y is \(y = 0.26x\).

Step 1: Identify the given variables

The given variables are \(x = 4.6\) and \(y = 1.2\).

Step 2: Use these variables to find the 'k' Value

You can find \(k\) by rearranging the formula to \(k = y/x\). Substituting the given values we get: \(k = 1.2 / 4.6\).

Step 3: Calculate 'k' Value

Performing the division from Step 2 gives \(k \approx 0.26\) (rounded to two decimal places).

Step 4: Write the Direct Variation Equation

Substitute the 'k' value into the formula \(y = kx\), which gives the final direct variation equation: \(y = 0.26x\).

Key Concepts

Variable Relationships

Understanding variable relationships is crucial when dealing with mathematical concepts, such as direct variation. In direct variation, two variables change in a consistent manner: when one variable increases, the other variable increases at a constant rate, and vice versa. This relationship can be represented by the equation y = kx, where y and x are the variables and k is the constant of variation.

In the given exercise, you're asked to identify how x and y are related. By establishing that for every unit increase in x, y increases by k units, you directly define their relationship. The discovery of this constant, k, is a crucial step for setting up the direct variation equation indicative of their relationship.

Proportional Reasoning

Proportional reasoning is a field in mathematics where relationships between quantities are expressed in ratios or fractions. When one quantity varies directly as another, the ratio between the two quantities remains constant. This concept helps in understanding how changes in one variable affect the other in a proportional manner.

In direct variation, the ratio of y to x is always equal to the constant k. In the exercise provided, once you calculate k as 0.26, you're employing proportional reasoning to understand that y is always 0.26 times as much as x, no matter the size of x. This ability to reason proportionally is instrumental in solving many real-world problems and is the backbone of direct variation.

Writing Equations

Writing equations is an essential skill in mathematics, as it allows us to express relationships and solve problems systematically. Once you understand the relationship between variables and can reason proportionally, writing the equation is the next logical step. An equation provides a precise mathematical model for the situation at hand.

In the context of the exercise, you're translating the direct variation between x and y into the equation y = kx. By substituting the calculated value of k (approximately 0.26), you're finalizing this model with y = 0.26x. This equation is now a tool that you can use to predict the value of y for any given x, as long as the direct variation relationship holds true. It's a concise representation of the pattern you've uncovered and serves as the basis for further analysis or application.