Problem 13 - Algebra 1: Concepts and Skills

Question

Find the constant of variation. \(y\) varies directly with \(x,\) and \(y=72\) when \(x=6\)

Step-by-Step Solution

Answer

The constant of variation \(k\) is 12.

Step 1: Write down the formula

Since \(y\) varies directly with \(x\), we use the formula \(y=kx\), where \(k\) is the constant of variation.

Step 2: Substitute the given values

Substitute the given values into the formula. We know that \(y=72\) and \(x=6\). So, replace \(y\) with 72 and \(x\) with 6 in the formula to get \(72=k*6\).

Step 3: Solve for \(k\)

To solve for \(k\), simply divide 72 by 6. This gives \(k=72/6=12\).

Key Concepts

Direct Variation

Direct variation describes a specific relationship between two variables where if one variable increases, the other does, too, by a constant proportion. This can be seen as a simple, straightforward relationship where quantity changes are consistently proportional. For example, if a variable \(y\) varies directly with variable \(x\), this means that when \(x\) changes, \(y\) changes in direct relation to \(x\).

In mathematical terms, this relationship is expressed through the formula \(y = kx\), where \(k\) is the constant of variation. This constant \(k\) is pivotal—it stays the same regardless of how \(x\) and \(y\) change. Understanding this allows you to determine one variable if you have the other variable and the constant. This principle has numerous applications across science and everyday calculations, demonstrating why direct variation is so important in mathematics.

Linear Equations

A linear equation is an algebraic expression that represents a line in a coordinate system. The simplest linear equation is of the form \(y = kx + b\), but in the case of direct variation, it simplifies to \(y = kx\) because there is no intercept (\(b = 0\)).

This equation represents a straight line passing through the origin (0,0), reflecting that the value of \(y\) depends solely on \(x\) and their direct proportion through the constant \(k\). Here's why linear equations are impactful:

• They are predictable and straightforward, making them fundamental in mathematics.
• They model relationships where changes are proportional and consistent, such as the exchange rate of currency or speed over time.

Learning to work with linear equations helps in grasping more complex algebraic concepts and forms the foundation for understanding many natural phenomena.

Mathematical Formulas

Mathematical formulas are like tools that help us solve problems. They encapsulate relationships and rules that make predicting numerical values easier. For direct variation, the fundamental formula is \(y = kx\), which is crucial for solving problems where one variable changes uniformly with another.

Using the given direct variation formula involves these steps:

• Identify which variables we are working with (in this case, \(x\) and \(y\)).
• Find the constant \(k\) using known quantities (as done in the provided solution by dividing \(y\) by \(x\)).
• Use \(k\) to predict or determine unknown values of \(x\) or \(y\).

Employing formulas simplifies solving equations, moving from complex-looking expressions to manageable calculations. Formulas provide a structured way to approach and solve mathematical problems easily and correctly.