Problem 103 - College Algebra with Modeling and Visualization

Question

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=\frac{1}{2},\) if \(y=\frac{3}{2}\) when \(x=\frac{2}{3}\)

Step-by-Step Solution

Answer

The value of \(y\) is \(\frac{9}{8}\).

Step 1: Understand the Problem

The problem states that variable \(y\) is directly proportional to variable \(x\). This means \(y = kx\), where \(k\) is a constant of proportionality. We need to find this constant first using the given values.

Step 2: Calculate the Constant of Proportionality

Using the known values \(y = \frac{3}{2}\) when \(x = \frac{2}{3}\), substitute these into the proportionality equation: \(\frac{3}{2} = k \cdot \frac{2}{3}\). To solve for \(k\), divide both sides by \(\frac{2}{3}\): \(k = \frac{3}{2} \div \frac{2}{3} = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4}\).

Step 3: Apply the Constant to Find y

Now that we have \(k = \frac{9}{4}\), substitute \(k\) into the proportionality equation to find \(y\) when \(x = \frac{1}{2}\). This gives \(y = \frac{9}{4} \times \frac{1}{2} = \frac{9}{8}\).

Step 4: Verify the Solution

Double check the solution by substituting \(y = \frac{9}{8}\) and \(x = \frac{1}{2}\) back into the equation \(y = kx\) using \(k = \frac{9}{4}\). We have \(\frac{9}{8} = \frac{9}{4} \times \frac{1}{2}\), which confirms the solution is correct.

Key Concepts

Constant of Proportionality

In direct variation problems, it is vital to grasp the concept of the *constant of proportionality*. This constant, often denoted by the letter "k", acts as a bridge connecting the two proportional quantities. When we say that "y is directly proportional to x", it translates into the mathematical relationship: \( y = kx \). Here, "k" links the changes in "y" to changes in "x" consistently.

• To find "k", use provided values of "y" and "x": \( y = \frac{3}{2} \) when \( x = \frac{2}{3} \).
• Substitute these into the proportionality equation: \( \frac{3}{2} = k \cdot \frac{2}{3} \).
• Solve for "k": \( k = \frac{3}{2} \div \frac{2}{3} = \frac{9}{4} \).

By understanding the constant of proportionality, you can easily determine how changes in one variable affect the other. This step is foundational in directly proportional relationships.

Solving Equations

Solving equations is a crucial skill in algebra that allows you to find unknown values by balancing both sides of an equation. When dealing with direct variation, after determining the constant of proportionality, the next step is applying it to find the unknown variable; in this case, "y".

• Given: \( k = \frac{9}{4} \) from our earlier calculations.
• To find "y" when \( x = \frac{1}{2} \), substitute into the equation: \( y = \frac{9}{4} \times \frac{1}{2} \).
• Solve the equation: \( y = \frac{9}{8} \).

By solving these simple linear equations, you can predict the behavior of directly proportional quantities under various conditions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to solve for desired variables. This skill is instrumental in transforming the original direct variation equation to isolate and find unknowns effectively.

• Initially, you have an equation \( \frac{3}{2} = k \cdot \frac{2}{3} \) to find "k".
• Manipulate this by dividing both sides by \( \frac{2}{3} \), which is equivalent to multiplying by its reciprocal \( \frac{3}{2} \), to isolate "k".
• Similarly, in the equation \( y = \frac{9}{4} \times \frac{1}{2} \), simplify by multiplying the constants.

Using algebraic manipulation, you ensure clarity and accuracy in problem-solving, allowing equations to be handled with ease and producing correct solutions.