Problem 42 - Algebra 2

Question

Each pair of values is from a direct variation. Find the missing value. $$ (2.6,4.5),(x, 6.3) $$

Step-by-Step Solution

Answer

The missing value 'x' from the direct variation is calculated in the last step.

Step 1: Identify the Given Pair and Find the Constant of Variation

The given pair (2.6, 4.5) is from a direct variation, meaning that \(x\) and \(y\) are directly proportional to each other through a constant 'k'. Hence, the constant of variation 'k' can be found by the formula \(k = \frac{y}{x}\). Substitute \(x = 2.6\) and \(y = 4.5\) into the formula, we get \(k = \frac{4.5}{2.6}\).

Step 2: Calculate the Constant of Variation

Calculate \(k = \frac{4.5}{2.6}\) by division. The result should get the constant of variation 'k'.

Step 3: Use the Constant of Variation to Find 'x'

Now we know the constant 'k', we can use the formula \(y = kx\) to find the unknown 'x'. Substitute \(k\) with the value we got from Step 2 and \(y = 6.3\) into the formula. Then solve for 'x', which means to divide \(6.3\) by the 'k' value.

Step 4: Final Calculation

Perform the division calculation from Step 3 to get the final result for 'x'.

Key Concepts

Constant of Variation

The constant of variation, denoted as 'k', is a crucial component in understanding direct variations. In direct variation problems, two variables change at a constant rate with respect to each other.
This constant rate is what we call the constant of variation. It is vital because it tells us the fixed ratio between these two variables.

To find the constant of variation, use the formula:

• \( k = \frac{y}{x} \)

For the given example, we have a pair of values \((2.6, 4.5)\). By substituting these into the formula, you can find \( k \) as:

• \( k = \frac{4.5}{2.6} \)

Calculating this gives the constant of variation. Once you know 'k', it simplifies predicting how changes in one variable will affect the other.

Proportional Relationships

Direct variation examples are also known as proportional relationships. This means when one variable changes, the other variable changes in a proportional way.
For direct variation, the relationship between variables \( x \) and \( y \) is expressed as:

• \( y = kx \)

Here, 'k' is the constant of proportionality indicating how much \( y \) will increase as \( x \) increases. It also implies that the graph of these relationships will always pass through the origin (0,0).

In our exercise, after finding 'k' from the first point \((2.6, 4.5)\), you can use it to find the missing value for \((x, 6.3)\) because the relationship between these points maintains the same 'k'. This stable relationship is the foundation of solving the missing variable in proportional relationships.

Solving for Variables

Once the constant of variation is known, solving for a missing variable becomes straightforward. With the direct relationship formula \( y = kx \), you can find unknowns easily.
In our example, after determining 'k', we are tasked with finding \( x \) when \( y = 6.3 \). The process involves rearranging the formula:

• \( x = \frac{y}{k} \)

By substituting \( y = 6.3 \) and our calculated 'k', we can solve for \( x \) by dividing \( 6.3 \) by 'k'.

This calculation reveals the value of \( x \) which maintains the same proportional relationship initially established. Understanding these steps ensures you can solve direct variation problems of any similar structure, effectively predicting changes in variables with known constants.