Problem 12 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=3, y=9$$

Step-by-Step Solution

Answer

The equation that relates the variables x and y that vary directly is \(y = 3x\).

Step 1: Determine the constant of variation

By substituting the values of x and y into the direct variation equation \(y = kx\), we can find the constant of variation. Given \(x = 3\) and \(y = 9\), the equation becomes \(9 = 3k\). We now solve for k.

Step 2: Solve for k

To solve for k, we divide both sides of the equation by 3. This gives us \(k = \frac{9}{3} = 3\). So the constant of variation k in this case is 3.

Step 3: Write the direct variation equation

Having identified k as 3, we substitute it back into the direct variation equation \(y = kx\). Thus, the equation that relates x and y for these variables that vary directly is \(y = 3x\).

Key Concepts

Constant of Variation

In algebra, when two variables are said to vary directly, this means they have a specific linear relationship. The constant of variation is the key element in this relationship. In direct variation, the equation is commonly expressed as \( y = kx \), where \( k \) is the constant of variation. This is the factor by which \( x \) is multiplied to find \( y \).

Let's break it down using an example. If you know that \( y = 9 \) when \( x = 3 \), you can determine \( k \) by dividing \( y \) by \( x \). This gives the equation \( k = \frac{y}{x} \), thus \( k = \frac{9}{3} = 3 \).

• The constant \( k \) determines how steep the line is when graphed.
• If \( k \) is positive, \( y \) increases with \( x \). If \( k \) is negative, \( y \) decreases with \( x \).

The constant of variation defines the strength and direction of the relationship between the variables.

Linear Equations

A linear equation is a statement of equality that involves two variable numbers. In the context of direct variation, a linear equation is represented as \( y = kx \). This equation shows a direct one-to-one relationship between \( x \) and \( y \). Each change in \( x \) results in a proportional change in \( y \).

Linear equations have several key characteristics:

• They graph as a straight line.
• The slope of the line is represented by \( k \), indicating how much \( y \) changes for a unit change in \( x \).
• They do not have variables raised to any power other than 1.

In our example \( y = 3x \), 3 is the slope, indicating that for every increase of 1 in \( x \), \( y \) increases by 3.

Algebraic Relationships

Algebraic relationships describe how different variables in an equation relate to each other. Direct variation is a specific type of algebraic relationship where one variable changes directly as another variable. This type of relationship is easy to understand and is fundamental in algebra due to its simplicity and direct proportionality.

In the equation \( y = kx \):

• Both \( x \) and \( y \) are directly proportional.
• The constant \( k \) ensures that this proportionality holds true across all values.
• This concept often applies to real-world situations, like speed and time or cost and quantity.

Understanding algebraic relationships like direct variation can help solve real-world problems by establishing clear relationships between variables, allowing predictions and comparisons effectively. The relationship between \( x \) and \( y \) in \( y = 3x \) provides an immediate understanding of how both variables will interact under various conditions.