Problem 1 - Algebra 2

Question

If \(y\) varies directly as \(x\) and \(y=18\) when \(x=15,\) find \(y\) when \(x=20\)

Step-by-Step Solution

Answer

\( y = 24 \) when \( x = 20 \).

Step 1: Understand Direct Variation

In direct variation, the relationship between two variables can be expressed by the equation \( y = kx \), where \( k \) is a constant. Our goal is to find \( k \) first, so we can use it to find \( y \) when \( x = 20 \).

Step 2: Find the Constant of Variation, \( k \)

We know from the problem that when \( x = 15 \), \( y = 18 \). Substitute these values into the direct variation formula, \( y = kx \):\[ 18 = k imes 15 \]To solve for \( k \), divide both sides by 15: \( k = \frac{18}{15} \). Simplifying gives \( k = \frac{6}{5} \).

Step 3: Use \( k \) to Find New \( y \)

Now that we have \( k = \frac{6}{5} \), use the direct variation formula again to find \( y \) when \( x = 20 \). Substitute \( k \) and \( x \) into \( y = kx \):\[ y = \frac{6}{5} imes 20 \].

Step 4: Calculate \( y \) for \( x = 20 \)

Compute the value of \( y \) by multiplying: \[ y = \frac{6}{5} imes 20 = 24 \]. The \( y \) value becomes 24 when \( x \) is 20.

Key Concepts

Constant of Variation

In the context of direct variation, the constant of variation is a particularly important concept. It is represented by the letter \( k \) in the direct variation equation \( y = kx \). The constant \( k \) helps to describe how changes in one variable affect the other in direct proportion. When a problem states that a relationship is one of direct variation, such as "\( y \) varies directly as \( x \)," it implies that for an increase or decrease in \( x \), \( y \) will increase or decrease in a consistent manner.

• To determine the constant of variation, you first need data values for both variables.
• Plug these values into the equation \( y = kx \).
• Solve for \( k \) by dividing the known value of \( y \) by the known value of \( x \).

Understanding the constant \( k \) is crucial for solving problems involving direct variation.

Relationship Between Variables

The relationship between variables in direct variation is quite simple and elegant. Here, the two variables are directly proportional. This means that as one variable increases, the other also increases at a constant rate governed by the constant of variation. The foundational equation for this relationship is \( y = kx \).
This implies several straightforward behaviors:

• If \( x \) is doubled, \( y \) will also double, assuming \( k \) remains constant.
• A smaller \( x \) results in a smaller \( y \).
• If \( x \) is zero, \( y \) will be zero because \( y = k \times 0 = 0 \).

It's this predictable pattern that allows such relationships to be easily calculated and represented graphically as straight lines passing through the origin in a coordinate system.

Solve for Constant

To solve for the constant of variation, \( k \), you need to follow a specific procedure that revolves around the direct variation formula \( y = kx \). In the given exercise, we have known values for both \( x \) and \( y \): \( y = 18 \) when \( x = 15 \).
Let's outline the steps to solve for \( k \):

• Start by inserting the known values into the equation: \( 18 = k \times 15 \).
• To isolate \( k \), divide both sides of the equation by 15: \( k = \frac{18}{15} \).
• Reduce the fraction to its simplest form: \( k = \frac{6}{5} \).

Once you've determined \( k \), you can use it to predict other values of \( y \) for different values of \( x \) using the same direct variation formula. This capability of solving for \( k \) is crucial for solving and understanding any problem involving direct relationships.