Problem 17 - Algebra 1: Concepts and Skills

Question

In Exercises 16–24, the variables x and y vary directly. Use the given values to write an equation that relates x and y. $$ x=7, y=35 $$

Step-by-Step Solution

Answer

The equation relating x and y through direct variation is \( y = 5x \).

Step 1: Understanding Direct Variation

In a direct variation equation, the variables x and y are related by the equation \( y = kx \), where k is the constant of variation. So, the first step is to identify the given values and input them into this equation.

Step 2: Determine the Constant of Variation (k)

Using the given values, x = 7 and y = 35, the constant of variation (k) can be found by rearranging the direct variation equation to isolate k: \( k = y / x \). So, substituting the given values: \( k = 35 / 7 = 5 \).

Step 3: Write the Equation of Direct Variation

Now that the constant of variation is known (k = 5), the equation that relates x and y can be written as: \( y = 5x \).

Key Concepts

Constant of Variation

In mathematics, the constant of variation is a specific value that relates two variables in a direct variation scenario. When two variables, such as \( x \) and \( y \), vary directly, they can be expressed by the equation \( y = kx \), where \( k \) is referred to as the constant of variation. This constant remains the same no matter which values of \( x \) and \( y \) you choose, provided they follow a direct variation relationship.
To find the constant of variation, you need to use the given values of \( x \) and \( y \) to solve for \( k \) using the formula \( k = \frac{y}{x} \). In the example provided, with \( x = 7 \) and \( y = 35 \), you would calculate it as:

• \( k = \frac{35}{7} \)
• \( k = 5 \)

Identifying this constant is crucial because it allows us to construct the direct variation equation, which describes how the two variables interact with one another.

Linear Equations

Linear equations are algebraic equations of the form \( y = mx + c \), where \( m \) and \( c \) are constants. These equations graph as straight lines on a coordinate plane. In direct variation scenarios, the linear equation takes the form \( y = kx \), where \( c = 0 \) and \( k \) is the constant of variation.
In the context of direct variation, the equation \( y = 5x \) is a linear equation where the line passes through the origin \((0, 0)\). This is because, in a direct variation, when \( x \) is zero, \( y \) is also zero. Thus, all direct variations are linear equations, making them easy to identify on a graph.
A few key features of linear equations include:

• The graph is a straight line.
• It has a constant rate of change, given by the slope \( k \).
• For direct variation, the line always passes through the origin.

Understanding linear equations helps in visualizing how changes in \( x \) affect \( y \), making it simpler to predict values and interpret relationships between variables.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In a direct variation relationship, an algebraic expression can describe how two quantities relate, showcasing the use of variables and constants strategically.

For example, in the direct variation equation \( y = 5x \), the expression involves:

• \( y \) and \( x \) as variables.
• \( 5 \) as a constant.

This algebraic expression indicates that for every unit increase in \( x \), \( y \) increases by a factor of 5. Algebraic expressions are powerful as they allow mathematicians to formulate and solve equations that describe real-world relationships in a concise manner.

The ability to manipulate and understand these expressions is essential for solving equations and understanding mathematical relationships. Students can practice simplifying and evaluating algebraic expressions to gain better problem-solving skills that are crucial for mastering algebra and beyond.