Problem 16 - Algebra Form and Function

Question

In Exercises 14-17, assume the two quantities are directly proportional to each other. If \(y=16\) when \(x=12\), find \(y\) when \(x\) is 9 .

Step-by-Step Solution

Answer

Question: If x and y are directly proportional, and when x = 12, y = 16, find the value of y when x = 9. Answer: When x = 9, the value of y is 12.

Step 1: Understanding Direct Proportionality

When two quantities are directly proportional, it means that as one increases or decreases, the other does the same. Mathematically, this relation can be expressed as: \(y = kx\), where \(k\) is the constant of proportionality.

Step 2: Calculate Constant of Proportionality

We know that when x = 12, y = 16. Therefore, we can use this information to find the constant of proportionality (k). From the formula of direct proportionality: \(y = kx => 16 = k * 12\) Now we will solve for k: \(k =\frac{16}{12} = \frac{4}{3}\)

Step 3: Find the Value of y When x = 9

Now that we have the constant of proportionality, we can use it to find the value of y when x = 9. Using the direct proportionality formula with the value of k we found and x = 9: \(y = kx = \frac{4}{3} * 9\)

Step 4: Calculate y

Multiplying \(\frac{4}{3}\) by 9, we obtain: \(y = \frac{4}{3} * 9 = 4 * 3 = 12\)

Step 5: Final Answer

When x is 9, the value of y is 12.

Key Concepts

Constant of Proportionality

When we talk about direct proportionality, we mean that two quantities change at the same rate. Imagine two friends who always save their money the same way, for every dollar one saves, the other saves too. This relationship can be described using a special number called the **constant of proportionality**. It assures that the ratio between the two quantities remains consistent. This constant is crucial because it allows us to determine unknown values once we know one of the quantities.

To find the constant of proportionality in a problem, like when we know that when one value is 16 and another is 12, we use the formula: \[ y = kx \] where \( y \) is 16 and \( x \) is 12. Solving for \( k \), we get:\[ k = \frac{y}{x} = \frac{16}{12} = \frac{4}{3} \] This constant \( \left( \frac{4}{3} \right) \) remains the same regardless of other values we might plug into the equation, giving us a dependable way to link \( y \) and \( x \) together.

Mathematical Formula

The concept of a **mathematical formula** is at the core of solving problems with direct proportionality. A formula acts as a roadmap, showing us the steps to follow when working with numbers. In cases of direct proportionality, our guide is:\[ y = kx \] This equation tells us that \( y \) is always equal to \( k \) multiplied by \( x \), where \( k \) is that constant value we discovered. Using this formula, we can solve for unknown values simply by plugging in what we know about \( x \) and using \( k \).

To find \( y \) when \( x \) is 9, knowing that \( k \) is \( \frac{4}{3} \), we plug these into our trusty formula:\[ y = \frac{4}{3} \times 9 = 12 \] This mathematical formula simplifies the problem-solving process because it provides a clear, repeatable structure to follow. It turns the abstract idea of proportionality into something concrete we can work with.

Algebraic Calculation

**Algebraic calculation** plays a vital role in finding solutions in directly proportional relationships. It's the tool that lets us handle the numbers logically and systematically. In our specific example, algebra helps us plug values into the formula and perform operations to find unknown quantities.

Calculating the constant of proportionality involved dividing to isolate \( k \):\[ k = \frac{y}{x} = \frac{16}{12} = \frac{4}{3} \] Next, we used algebraic multiplication to find \( y \) when \( x \) was 9:\[ y = k \cdot x = \frac{4}{3} \cdot 9 \] Breaking it down, we multiply \( 9 \) by \( \frac{4}{3} \), which gives:\[ y = 12 \] Algebraic calculation doesn't just involve numbers; it involves understanding what each number represents and how it relates to others. These operations form the backbone of our problem-solving strategy. By logically progressing from one step to the next, algebra makes it easier to follow a path from known information to new insights.