Problem 11 - Intermediate Algebra

Question

Find the constant of variation for each of the stated conditions. \(y\) varies directly as \(x\), and \(y=8\) when \(x=12\).

Step-by-Step Solution

Answer

The constant of variation is \( \frac{2}{3} \).

Step 1: Understand Direct Variation

In a direct variation, the relationship between two variables can be expressed by the equation \( y = kx \), where \( k \) is the constant of variation.

Step 2: Substitute the Given Values

Substitute the given values into the equation. We have \( y = 8 \) and \( x = 12 \). So, the equation becomes \( 8 = k \times 12 \).

Step 3: Solve for the Constant of Variation

To find \( k \), divide both sides of the equation by \( 12 \). This gives \( k = \frac{8}{12} \).

Step 4: Simplify the Expression

Simplify \( \frac{8}{12} \) by dividing both numerator and denominator by their greatest common divisor, which is \( 4 \). So, \( k = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \).

Key Concepts

Understanding the Constant of Variation

In algebra, the concept of direct variation is key to understanding how two variables interact with each other. When we say that a variable "varies directly" with another, it means there is a consistent relationship. This relationship is captured by the constant of variation, denoted as \( k \). For the equation of direct variation \( y = kx \), \( k \) dictates how much \( y \) will change as \( x \) changes. This constant essentially tells us the rate of change or how steep the line is if we were to graph the equation.

• If \( k \) is positive, both variables increase together.
• If \( k \) is negative, one variable decreases as the other increases.
• A larger absolute value of \( k \) indicates a steeper or quicker rate of change.

In the context of real-world problems, knowing \( k \) helps make predictions or understand scenarios where proportional relationships are involved.

Exploring Linear Equations

Linear equations form the backbone of direct variation, especially in algebraic terms. A linear equation is any equation that forms a straight line when graphed on a coordinate plane. The basic form is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
However, in direct variation, the equation simplifies to \( y = kx \) since the line passes through the origin, meaning there is no y-intercept (or \( b = 0 \)). This makes direct variation a special type of linear equation. Understanding linear equations helps us easily interpret direct variations. When graphed, the line will pass through the origin (0,0) and slope upwards or downwards depending on whether \( k \) is positive or negative.Grasping these principles can significantly ease the process of analyzing both direct variation and other kinds of linear relationships in mathematics.

Unlocking Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( x \) and \( y \)), and operations (such as addition, subtraction, multiplication, and division).
They are crucial for expressing relationships in math, whether it be equations for direct variation, like \( y = kx \), or more complex formulas. In dealing with algebraic expressions, it's important to:

• Understand the variables and constants in play.
• Use substitution to simplify expressions or solve equations, as shown in the step where values were substituted to find \( k \).
• Perform operations like simplifying fractions, factoring, or expanding expressions as required.

By mastering these skills, you can easily manipulate equations to find unknowns, predict values, and make mathematical connections clear. This forms a foundational necessity for tackling more advanced algebraic concepts.