Problem 3 - Algebra A Combined Function

Question

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=6\) when \(x=4\)

Step-by-Step Solution

Answer

The constant of variation is 1.5 and the equation is \(y = 1.5x\).

Step 1: Understand Direct Variation

When two variables, say \(y\) and \(x\), are related by direct variation, it means \(y\) is equal to a constant \(k\) times \(x\), or \(y = kx\). This also means if \(y\) increases, \(x\) increases proportionally and vice versa.

Step 2: Identify Given Values

You're given that \(y = 6\) and \(x = 4\). We will use these values to find the constant \(k\) in the direct variation equation \(y = kx\).

Step 3: Solve for the Constant of Variation \(k\)

Substitute the given values into the equation of direct variation. Plug \(y = 6\) and \(x = 4\) into \(y = kx\). This gives \(6 = k \times 4\).

Step 4: Calculate \(k\)

Solve the equation \(6 = 4k\) for \(k\). Divide both sides by 4 to isolate \(k\). This results in \(k = \frac{6}{4} = 1.5\).

Step 5: Write the Direct Variation Equation

Now that we have \(k = 1.5\), substitute \(k\) back into the direct variation formula \(y = kx\) to get \(y = 1.5x\). This is the direct variation equation.

Key Concepts

Constant of Variation

The constant of variation, often denoted as "\(k\)", is a fundamental component in understanding direct variation. It represents the factor by which the independent variable, \(x\), is multiplied to achieve the dependent variable, \(y\). This constant does not change for the given relationship and defines how \(y\) scales with \(x\).

• To find the constant of variation, you simply rearrange the direct variation formula \(y = kx\) to solve for \(k\).
• Using data provided, such as \(y = 6\) when \(x = 4\), substitute these values into the formula: \(6 = 4k\).
• Then solve for \(k\) by dividing both sides by the coefficient of \(k\), giving \(k = \frac{6}{4} = 1.5\).

This step reveals the exact multiplier, \(k\), that connects any \(x\) value with its corresponding \(y\) value.

Variations in Algebra

Variations in algebra can involve different types of relationships between variables, with direct variation being one of the simplest forms. In a direct variation, as one variable increases or decreases, the other does the same. This linear relationship is straightforward and easiest to recognize when expressed in the form \(y = kx\).

• Direct variation has a single constant, \(k\), which is critical in defining the relationship between variables.
• The relationship is proportional, making it easy to predict changes in \(y\) based on changes in \(x\).
• To identify a direct variation, look for the characteristic that doubling \(x\) will double \(y\), and halving \(x\) will halve \(y\), maintaining a consistent ratio.

Understanding these variations provides clarity in solving many algebraic equations and real-world problems.

Linear Relationships

Linear relationships are a pervasive concept in both mathematics and real-world applications. In algebra, direct variation is a prime example of a linear relationship because the graph of \(y = kx\) results in a straight line through the origin.

• In these relationships, there are no exponents or powers, which means the graph will always be a straight line.
• Because the graph includes both \(x\) and \(y\), as one of these variables changes, the other shifts in direct proportion to the constant \(k\).
• These relationships are visualized as consistent and predictable, with the slope of the line equating to the constant of variation \(k\).

Linear relationships provide a clear and concise representation of direct variations, aiding in the understanding and application of these principles across various contexts.