Problem 48 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The missing value \(x\) is approximately \(1.857\).

Step 1: Identify the formula for inverse variation

The formula for inverse variation is \(xy = k\). Here, \(x\) and \(y\) are variables and \(k\) is the constant of variation.

Step 2: Use the given pair to calculate the constant of variation

Use the first pair of given values (2.6, 4.5) in our inverse variation formula. Hence, \(2.6*4.5=k\). This will provide us with the value of our constant \(k\).

Step 3: Calculate the constant of variation

Calculate \(k\) by multiplying \(2.6\) and \(4.5\) which equals \(11.7\). Hence, \(k=11.7\).

Step 4: Use the constant to find the missing variable

Now, use \(k\) in the formula with the second pair, where \(y = 6.3\), to find \(x\). This gives us the equation \(x*6.3=11.7\).

Step 5: Solve for the missing variable

Solve for \(x\) in the equation \(x*6.3=11.7\) by dividing both sides by \(6.3\). This gives us the value of \(x\).

Key Concepts

Understanding the Constant of Variation

In an inverse variation relationship, the constant of variation is crucial. This constant, often denoted as \(k\), tells us how the two variables, \(x\) and \(y\), relate to each other. In simpler terms, the product of the two variables is always equal to this constant value, ensuring that as one variable increases, the other decreases proportionally. For example, if we have a pair of values \((2.6, 4.5)\) in an inverse variation, we use these values in the equation \(xy = k\). This means

• \(2.6 \times 4.5 = k\)
• \(k = 11.7\)

The constant of variation does not change, it remains 11.7 for all pairs of \(x\) and \(y\) that follow this specific inverse variation relationship. This constancy helps us solve for missing values when other instances of \(x\) or \(y\) are presented.

Solving for the Missing Variable

When given part of a pair in inverse variation, finding the other value requires using the constant of variation. In our scenario, we had the second pair \((x, 6.3)\) incomplete. But since we've determined \(k\) is 11.7 from our earlier steps, the equation becomes:

• \(x \times 6.3 = 11.7\)

To solve for \(x\), we need to isolate it. This involves performing the opposite operation of multiplication, which is division. Divide both sides by 6.3 to get:

• \(x = \frac{11.7}{6.3}\)
• \(x = 1.857\)

The answer we find is approximately 1.857, giving us the missing part of our pair. The step of isolating the variable by performing algebraic operations reiterates core concepts crucial in algebra: balancing the equation by performing opposite operations.

Basics of Solving Algebraic Equations

Solving algebraic equations is a key skill in understanding mathematical relationships like inverse variation. Algebra involves finding the unknown values by manipulating equations using different operations—and it always aims to maintain balance. In the example above, solving for \(x\) involved:

• Recognizing what operation is being performed (multiplication of \(x\) by 6.3).
• Identifying the inverse operation (division) to retrieve the unknown \(x\).
• Applying the operation equitably to both sides of the equation to preserve equality.

This fundamental approach ensures that whenever you need to solve for an unknown in any mathematical context, you can confidently handle the equation, no matter how complex it seems. Remember, keeping track of operations and how they affect both sides of an equation is at the heart of successful problem-solving in algebra.