Problem 37 - A Graphical Approach to Precalculus with Limits

Question

find the constant of variation \(k\) and the undetermined value in the table if \(y\) is directly proportional to \(x\). $$\begin{array}{|c|c|c|c|c|} \hline x & 3 & 5 & 6 & 8 \\ \hline y & 7.5 & 12.5 & 15 & ? \end{array}$$

Step-by-Step Solution

Answer

The constant of variation \(k\) is 2.5, and the missing \(y\) value is 20.

Step 1: Understand Direct Variation

Direct variation means that two variables, \(x\) and \(y\), have a constant ratio. This can be written as \(y = kx\) where \(k\) is the constant of variation. Our task is to identify \(k\) using known values and then find the missing \(y\).

Step 2: Calculate the Constant of Variation \(k\)

We have a pair of \(x\) and \(y\) values: for \(x = 3\), \(y = 7.5\). Substituting these values into the equation \(y = kx\) gives us \(7.5 = k \cdot 3\). Divide both sides by 3 to solve for \(k\): \(k = \frac{7.5}{3} = 2.5\).

Step 3: Use \(k\) to Find Missing \(y\) Value

Now that we have \(k = 2.5\), substitute \(x = 8\) into the equation \(y = kx\). This gives \(y = 2.5 \times 8\). Calculating gives \(y = 20\).

Step 4: Verify Consistency (Optional)

To ensure our solution is consistent, we can check other values: for example, when \(x = 5\), \(y\) could be \(2.5 \times 5 = 12.5\) which matches the table. For \(x = 6\), \(y\) would be \(2.5 \times 6 = 15\), verifying consistency with all given values.

Key Concepts

Constant of Variation

The constant of variation, often denoted by the symbol \(k\), plays a crucial role in equations describing direct variation. In this context, we say that \(y\) is directly proportional to \(x\) if there exists a non-zero constant \(k\) such that \(y = kx\). This equation illustrates the direct variation relationship.When approaching problems involving direct variation, identifying the constant of variation is a vital first step. Here's how you can find it:

• Choose a pair of \(x\) and \(y\) values from your data set.
• Substitute these values into the equation \(y = kx\).
• Solve for \(k\) by rearranging the equation to \(k = \frac{y}{x}\).

Once \(k\) is calculated, it remains constant for all pairs in the dataset that conform to this proportionality.

Proportional Relationship

A proportional relationship signifies that two variables maintain constant ratios. That means the ratio \(\frac{y}{x}\) is consistent for all pairs within a set. This relationship underscores the fundamental principle of direct variation, where any change in one variable is reflected proportionally in the other.To recognize a proportional relationship, follow these indicators:

• Each increase or decrease in \(x\) results in a corresponding linear change in \(y\).
• The graph of this relationship is a straight line passing through the origin (0,0).
• Every pair of \(x\) and \(y\) will yield the same \(k\), verifying the consistency over the dataset.

By checking the constancy of the ratio or confirming that a straight line graph passes through the origin, you verify the existence of a proportional relationship.

Linear Equation

Linear equations are foundational in understanding relationships between variables, especially in the context of direct variation. A linear equation in two variables \(x\) and \(y\) can be expressed as \(y = mx + b\). In direct variation, however, the equation simplifies to \(y = kx\) since the line always passes through the origin, eliminating the constant term \(b\).In analyzing a linear equation for direct variation:

• The slope \(m\) in a general linear equation is equivalent to the constant of variation \(k\) in direct variation scenarios.
• This slope describes how steep the line is, indicating how much \(y\) changes with a unit change in \(x\).
• A direct variation is a particular type of linear equation where any line goes through the point (0,0).

These qualities of linear equations assist in understanding and solving problems related to proportional relationships in real-world contexts.