Problem 15 - College Algebra

Question

Determine the constant of variation for each stated condition. \(y\) varies directly as \(x^{2},\) and \(y=45\) when \(x=3\)

Step-by-Step Solution

Answer

The constant `k` of variation is 5.

Step 1: Write down the direct variation formula

The direct variation formula when the variation is happening with reference to a square is given as \(y=kx^2\), where \(k\) is the constant of variation that we need to find.

Step 2: Substitute given values of x and y into the formula

From the problem, we have \(y = 45\) when \(x = 3\). Substituting these values into the equation from step 1 results in: \(45 = k(3^2)\).

Step 3: Solve the Equation for k

To solve for \(k\), divide both sides of the equation by \(3^2 = 9\): \( \frac{45}{9} = k\), which simplifies to: \(k = 5\).

Step 4: Verification

To verify our answer, we can substitute \(k = 5\) into the original variation equation and check if it holds true for \(y = 45\) when \(x = 3\). Doing so, we get: \(y = 5*(3^2)\) which simplifies to \(y = 45\).

Key Concepts

Understanding the Constant of Variation

In mathematics, the constant of variation, denoted as "\(k\)", plays a vital role in direct variation equations. Direct variation represents a relationship where one variable changes consistently with respect to another variable, often indicated as \(y = kx^n\) for various powers of \(x\).

• **Direct Relationship:** This means that if one variable increases or decreases, the other does so too, following a proportional pattern.
• **Constant \(k\):** It's a fixed number that indicates how much \(y\) will change when \(x\) changes by a certain factor.

In the exercise, \(k\) helps us understand how \(y\) is directly proportional to the square of \(x\). Thus, the determination of \(k\) provides the precise mathematical expression of this direct relationship. Once you grasp how to find \(k\), the rest of the problem becomes much clearer.

Algebraic Equations in Direct Variation

An algebraic equation is an equation involving variables and constants, often presented with operators like addition, subtraction, multiplication, or division. In direct variation cases like our exercise's formula \(y = kx^2\), this equation helps establish that relationship.

• **Structure:** Typically follows the form \(y = kx^n\).
• **Variables & Constants:** Here, \(y\) and \(x\) are variables, while \(k\) is a constant.

Understanding algebraic equations allows students to represent complex relationships with simplified expressions. In direct variation, the entire equation itself means that the change in \(y\) directly depends on \(x\)'s square when \(k\) is known. By substituting given numbers into the equation, we can solve real-world problems or theoretical exercises.

Step-by-Step Guide to Solving Equations for Direct Variation

Solving equations is a foundational mathematical technique used to isolate a variable and determine its value. Here's how you can solve an equation concerning the direct variation to find the constant of variation \(k\):

• **Identify the Formula:** Begin with the direct variation form relevant to your problem, such as \(y = kx^2\).
• **Substitute Known Values:** Next, plug in the given values of \(x\) and \(y\) into the formula.
• **Perform Arithmetic Operations:** Simplify the resulting equation by performing arithmetic operations to isolate \(k\). For example, dividing both sides by \(x^n\).
• **Verification:** Check your solution by substituting \(k\) back into the original equation to ensure consistency with given values.

Following these steps in order ensures clarity and correctness in solving equations, making the process less intimidating and more accessible. Practice with various problems to strengthen your equation-solving skills.