Problem 39 - Algebra 1

Question

Which equation models the ratio form of direct variation? $$\begin{array}{llll}\text { (A) } \frac{-4}{3}=\frac{y}{x} & \text { (B) }-3 y=4 x-1 & \text { (C) } \frac{4}{-3}=x+y &\text { (D) } y=-\frac{4}{3} x+6\end{array}$$

Step-by-Step Solution

Answer

The correct answer is Option (A): \(\frac{-4}{3}=\frac{y}{x}\) which can be rewritten as \(y=\frac{-4}{3}x\). This is the only choice that represents the ratio form of direct variation.

Step 1: Understand Direct Variation Formula

The formula for direct variation is \(y=kx\), where \('k'\) is the constant of variation. Therefore, in this exercise, look for an equation that can be written in the form \(y=kx\).

Step 2: Explore Option (A)

\(\frac{-4}{3}=\frac{y}{x}\) can be rewritten as \(y=\frac{-4}{3}x\) which is in the form of \(y=kx\), therefore is a ratio of direct variation.

Step 3: Explore Option (B)

-3y=4x-1 cannot be rewritten as \(y=kx\) because there is a subtraction factor (-1). Therefore, this is not in the ratio form of direct variation.

Step 4: Explore Option (C)

\(\frac{4}{-3}=x+y\) cannot be rewritten as \(y=-\frac{4}{3}x\) because there is an addition factor (y). Therefore, this is not in the ratio form of direct variation.

Step 5: Explore Option (D)

y=-\(\frac{4}{3}\)x+6 cannot be rewritten as \(y=kx\) because there is an addition factor (+6). Therefore, this is not in the ratio form of direct variation.

Key Concepts

Constant of Variation

The constant of variation is a crucial element in direct variation equations. It is usually represented by the letter "\( k \)". In a direct variation relationship, the "\( y \)" values change consistently in proportion to changes in "\( x \)" values, thanks to this constant. It reflects the constant rate at which "\( y \)" changes with "\( x \)".

For example, consider the direct variation equation \( y = kx \). Here, if \( k = \frac{-4}{3} \), it means that for every unit increase in \( x \), \( y \) decreases by \( \frac{4}{3} \). This consistent rate of change is what makes \( k \) so important in understanding how the two variables relate. Knowing \( k \) allows us to quickly calculate one variable when given the other in the context of direct variation.

Direct Variation Formula

Direct variation is described by the simple equation \( y = kx \), where "\( y \)" varies directly with "\( x \)" and \( k \) is known as the constant of variation. This formula is foundational for identifying direct variation relationships.

In practical terms, whenever you see an equation of the form \( y = kx \), you can say "\( y \) varies directly with \( x \)". Direct variation implies a predictable and linear relationship between \( x \) and \( y \).

• "\( k \)" must be a non-zero constant.
• The graph of a direct variation equation is a straight line through the origin (0,0).

Recognizing this formula in a problem allows you to quickly identify and solve direct variation problems by relating \( x \) and \( y \).

Equation Transformation

Understanding how to manipulate equations is vital when identifying direct variation among multiple equation forms. The goal with direct variation equations is to transform them into the \( y = kx \) form, if possible.

For example, consider the equation \( \frac{-4}{3} = \frac{y}{x} \). By multiplying both sides by \( x \), you transform the equation into \( y = \frac{-4}{3}x \). This conversion highlights the constant of variation and aligns with the direct variation formula.

• Look for opportunities to isolate "\( y \)" and "\( x \)" on opposite sides of the equation.
• Check for additional terms (e.g., constants, other variables) that may prevent the equation from being a direct variation.

Mastering equation transformation is crucial for navigating complex equations and extracting a direct variation relationship when present.