Problem 41 - Algebra 2

Question

Each pair of values is from a direct variation. Find the missing value. $$ (8.3,7.1),(5, y) $$

Step-by-Step Solution

Answer

Solving the equation gives \(y \approx 4.3\). Thus the missing value is approximately 4.3

Step 1: Identify a proportion

As this is a direct variation, the ratios of the values in each pair must be equal. So you can write the equation \(\frac{8.3}{7.1} = \frac{5}{y}\)

Step 2: Solve the proportion

To solve the proportion for y, you first need to cross multiply. Doing so provides the equation \(8.3 * y = 7.1 * 5\)

Step 3: Find the missing value

To isolate y, divide both sides of the equation by 8.3. This gives \(y = \frac{7.1*5}{8.3}\). Solving this equation gives the numerical value for y

Key Concepts

Understanding Proportion

A proportion is an equation that states that two ratios are equal. It plays a crucial role in understanding direct variation problems like the one in the exercise. In direct variation, if you have two pairs of related quantities, their ratios will always be the same. For example, if you have two quantities, A and B, that vary directly, they can be expressed as a proportion:

• \( \frac{A_1}{B_1} = \frac{A_2}{B_2} \)

This means that the relationship between the first pair of quantities is the same as the relationship between the second pair. The use of proportion allows us to find unknown values when one of the pairs is incomplete, by maintaining the equality of ratios.

The Technique of Cross Multiplication

Cross multiplication is a handy technique used to solve proportions. It involves multiplying across the terms of the ratios to eliminate the fractions.

• Imagine you have a proportion: \( \frac{a}{b} = \frac{c}{d} \)
• You can "cross multiply" this proportion by multiplying the means and extremes to get: \( a \times d = b \times c \)

This step transforms the original proportion into an equation without any fractions, making it easier to solve. In our exercise, when we set up the equation \( \frac{8.3}{7.1} = \frac{5}{y} \), cross multiplication helps us move from dealing with two fractions to a straightforward multiplication equation: \( 8.3 \times y = 7.1 \times 5 \). This greatly simplifies finding the missing variable.

Solving Equations to Find the Missing Variable

Once we've used cross multiplication to simplify our proportion to a simple equation, the next step is to solve for the unknown. Solving an equation involves isolating the variable you want to find on one side of the equation. In the given problem, after cross multiplication, we get:

• \( 8.3 \times y = 7.1 \times 5 \)

To find \( y \), you need to isolate it by dividing both sides of the equation by 8.3:

• \( y = \frac{7.1 \times 5}{8.3} \)

Calculating this division will provide the exact value of \( y \), completing the solution. Solving such equations requires an understanding of basic arithmetic operations: multiplication, division, and simplifying fractions, ensuring you maintain equality throughout the process.