Problem 63 - Algebra 2

Question

Determine whether \(y\) varies directly with \(x .\) If so, find the constant of variation. $$ y-2=2 x $$

Step-by-Step Solution

Answer

No, y does not vary directly with x because the equation cannot be expressed in the form y = kx due to the additional +2.

Step 1: Rearrange the equation to the direct variation form

Determine if the given equation can be expressed in the form of direct variation, which is ｙ = kx, where k is the constant of variation. Start by isolating ｙ on one side of the equation.

Step 2: Solve for y

Add 2 to both sides of the equation to isolate ｙ. The equation becomes y = 2x + 2.

Step 3: Analyze the resulting equation

Compare the resulting equation with the standard direct variation equation. If y varies directly with x, the equation would be in the form y = kx, where k is a constant. In this case, y = 2x + 2 does not match the form y = kx because of the additional +2. Therefore, y does not vary directly with x.

Key Concepts

Constant of Variation

When we discuss direct variation, we're talking about a relationship between two variables where one is a constant multiple of the other. This constant multiple is known as the constant of variation, often represented as 'k' in equations. For an equation to represent direct variation, it must be possible to write it in the form ( y = kx ), where 'y' is the dependent variable, 'x' is the independent variable, and 'k' is the constant of variation. The value of 'k' determines how strongly 'y' depends on 'x': the larger the 'k', the more 'y' changes in response to changes in 'x'.

In the provided exercise, the student needs to identify if there's a direct variation and, if so, what the constant of variation is. The original equation, ( y-2=2x ), seems close, but it includes a (-2) term that tells us something additional is at play, making it more than simply 'y' being a multiple of 'x'. Through proper algebraic manipulation, determining whether 'k' exists in this equation and if 'y' varies directly with 'x' is critical.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them clearer or to solve for a particular variable. Essential techniques include adding, subtracting, multiplying or dividing both sides of an equation by the same amount; distributing a constant over a parenthesis; combining like terms; and factoring.

The goal of algebraic manipulation in the context of direct variation is to isolate the variable 'y' and express it in terms of 'x', ideally resulting in the form mentioned earlier: ( y = kx ). In the exercise, the student begins by adding 2 to both sides of the initial equation to isolate 'y'. This is a classic algebraic manipulation move and is essential to comparing the resulting equation with the standard form of direct variation. ( y = 2x + 2 ) is the manipulated equation. However, the '+2' at the end indicates an extra step or term - unlike what a direct variation formula would include.

Equation Solving

Equation solving is finding the value of the variables that make an equation true. It's a fundamental part of algebra and is incredibly useful across all areas of mathematics. To solve an equation, you perform a series of algebraic manipulations, which usually involves isolating the variable of interest. In the case of direct variation, we want to isolate 'y' so that its relationship with 'x' is explicit and follows the form ( y = kx )

In our specific exercise, the student aimed to solve for 'y' to determine whether it directly varies with 'x'. After algebraic manipulation, they were left with ( y = 2x + 2 ). This tells us that while 'y' can be expressed as a function of 'x', the equation does not represent direct variation because of the +2 term. Therefore, the step by step solution concludes that 'y' does not vary directly with 'x', and thus, there is no constant of variation in this context. This example emphasizes the importance of equation solving in identifying direct variation and differentiating it from other types of relationships between variables.