Problem 57 - Algebra 2

Question

For each direct variation, find the value of \(y\) when \(x=2\) $$ y=1 \text { when } x=4 $$

Step-by-Step Solution

Answer

\(y = \frac{1}{2}\) when \(x = 2\) in the direct variation.

Step 1: Determine the Constant of Variation

Given that the direct variation can be described by the equation of the form \(y = kx\), where \(k\) is the constant of variation, we can find \(k\) by substituting the given values of \(x\) and \(y\): \(y=1\) when \(x=4\). So, the equation for the constant of variation is \(1 = k \cdot 4\). To find \(k\), divide both sides by 4: \(k = \frac{1}{4}\).

Step 2: Write the Direct Variation Equation

Now that we have the constant of variation \(k = \frac{1}{4}\), we can write the direct variation equation as \(y = \frac{1}{4}x\).

Step 3: Find the value of \(y\) when \(x=2\)

To find the value of \(y\) when \(x=2\), substitute \(2\) for \(x\) in the direct variation equation: \(y = \frac{1}{4}\cdot 2 = \frac{1}{2}\).

Key Concepts

Constant of Variation

In a direct variation, the constant of variation is a key concept that stipulates how one variable changes in proportion to another. For every direct variation equation of the form \(y = kx\), where \(y\) and \(x\) are variables, the constant of variation, represented by \(k\), signifies the consistent rate of change between \(y\) and \(x\).

Understanding this constant is essential because it determines the slope of the line in a graph representing the relationship between \(y\) and \(x\). To determine the constant of variation, we divide the value of one variable by the corresponding value of the other variable when both are known. This ratio remains constant for all values of \(x\) and \(y\) involved in this type of relationship.

In the provided exercise, the initial value \(y=1\) when \(x=4\) allows us to calculate the constant of variation as \(k = \frac{1}{4}\). This tells us that for every unit increase in \(x\), \(y\) increases by \(\frac{1}{4}\) of that unit.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In the context of direct variation, the algebraic expression manifests itself as the formula \(y = kx\), reflecting the relationship between variables \(x\) and \(y\).

The process of forming an algebraic expression involves understanding the relationship between variables and representing this relationship concisely with symbols. In a direct variation scenario, the constant of variation \(k\) connects the variables in a simple yet profound way, defining how each variable's change is dependent on the other.

Creating and simplifying algebraic expressions are fundamental skills in algebra. For instance, once we've found the constant of variation \(k\), we use it to form the specific algebraic expression that will enable us to solve for unknown variable values, like determining \(y\) when given a new value for \(x\).

Solving Equations

Solving equations is a crucial skill in algebra, which involves finding the values of variables that make the equation true. The process usually entails a series of steps where we manipulate the equation by performing arithmetic operations to isolate the variable of interest.

When we talk about direct variation problems, after establishing the direct variation equation \(y = kx\), our next task is to 'solve the equation,' meaning to find the value of one variable, typically \(y\), for a given value of the other variable, \(x\).

In our example, with the direct variation equation \(y = \frac{1}{4}x\) and \(x=2\), we substitute \(2\) into the equation in place of \(x\) and solve for \(y\): \(y = \frac{1}{4} \cdot 2\), which simplifies to \(y = \frac{1}{2}\). This step is the very essence of solving the equation: we transformed an algebraic expression with an unknown \(y\) into a numerical value that gives us specific information about the relationship between \(x\) and \(y\) in the context of direct variation.