Problem 46 - Algebra 2

Question

Each pair of values is from an inverse variation. Find the missing value. $$ (9,5),(x, 3) $$

Step-by-Step Solution

Answer

The missing value \(x\) in the pair \((x, 3)\) is 15.

Step 1: Understand Inverse Variation

Inverse variation means the product of the two variables \(x\) and \(y\) is constant. This can be represented by the formula \(xy = k\), where \(k\) is the constant.

Step 2: Identify the given pairs

The given pairs are (9, 5) and \((x, 3)\). Pair (9, 5) can be used to calculate the constant of variation \(k\).

Step 3: Compute the Constant of Variation

Calculate the constant \(k\) from the first pair by multiplying the two values together. So \(k = 9 * 5 = 45\).

Step 4: Find the Missing Value

Substitute the value of \(k\) and \(y\) from the second pair into the inverse variation formula to find \(x\). So, \(k = xy\) becomes \(45 = x * 3\). Solving for \(x\), we get \(x = 45 / 3 = 15\).

Key Concepts

Constant of Variation

The constant of variation is a crucial part of understanding the relationship between two variables in an inverse variation setup. When dealing with inverse variations, the product of the two involving variables remains unchanged. This product is referred to as the constant of variation, denoted by the symbol \(k\). In any inverse variation, the relationship can be expressed using the formula \(xy = k\), where \(x\) and \(y\) are the variables. The concept is simple.

• If you increase one variable, the other must decrease to maintain the constant product \(k\).
• Conversely, if you decrease one, the other must increase.

To determine \(k\), you only need one pair of values \((x, y)\). Multiply them, and you will have your constant. Once found, this constant helps to explore other unknown variables in the situation, should the relationship type remain the same.

Algebraic Expressions

Algebraic expressions are fundamental tools in solving problems involving variations, whether direct or inverse. In an inverse variation problem, the expression \(xy = k\) is a simple algebraic representation of the relationship between \(x\) and \(y\). Such expressions involve variables, constants, and various arithmetic operations. In our specific problem scenario, we derive the expression from the given data:

• The pair \((9, 5)\) leads to the algebraic expression \(9 \times 5 = k\), finding \(k = 45\).
• This expression is key as it encapsulates the entire variation relationship.

By manipulating this expression, you can solve for unknowns, always returning to the same expression format to ensure consistency in the relationship you are analyzing.

Solving Equations

Solving equations in the context of inverse variation involves isolating the unknown variable. When you know the constant of variation \(k\), you can use the given formula \(xy = k\) to find missing values. Here's how you solve it:

• Start by substituting the known values into the equation.
• In our exercise, \(k = 45\) and \(y = 3\) for the second pair.

The equation becomes \(45 = x \times 3\).
To solve for \(x\), rearrange the equation by dividing both sides by 3:

• \(x = 45/3\)
• Calculating gives \(x = 15\)

The solution is simple arithmetic once the equation is appropriately set up. This process of rearranging and simplifying is the heart of solving inverse variation problems, ensuring you find the correct value for unknowns consistently.