Problem 39 - Algebra 2

Question

Find the constant of variation for a direct variation that includes the given values. \((-1,7)\)

Step-by-Step Solution

Answer

The constant of variation k for the direct variation that includes the point (-1, 7) is -7.

Step 1: Understanding the Problem

In a direct variation, the relationship between two variables can be described by the equation y = kx, where k is the constant of variation. We are given a point (-1, 7) which satisfies this equation, and we need to find the value of the constant k.

Step 2: Substitute the Given Values into the Direct Variation Equation

Substitute the x and y values from the given point into the direct variation equation (y = kx). We have y = 7 and x = -1, which gives us 7 = k(-1).

Step 3: Solving for the Constant of Variation

To find k, divide both sides of the equation by -1 to isolate k. This gives us k = 7 / (-1), which simplifies to k = -7.

Key Concepts

Direct Variation

Direct variation is a fundamental concept in algebra that describes a linear relationship between two variables. When one variable is a constant multiple of the other, we say they vary directly. The classic form of the direct variation equation is \( y = kx \), where \( k \) is known as the constant of variation. This constant remains unchanged and defines the strength and nature of the relationship. For example, if \( y \) is directly proportional to \( x \) with a constant of variation of 3, then as \( x \) increases, \( y \) will also increase, and will always be three times as large as \( x \).

In the given exercise, identifying the direct variation relationship is crucial. It's like unravelling a puzzle to find out how closely two quantities are dancing together; in this case, the steps provided help us uncover the rhythm of their movements.

Algebraic Equations

Algebraic equations are the bread and butter of algebra — they are mathematical statements that express the equality between two expressions. These equations can involve numbers, variables, or both, and they lay down the rules for how the variables interact with each other. To solve an algebraic equation is to find the value of the variable that makes the equation true.

Our given example involves a simple algebraic equation derived from a direct variation. Here, the equation takes the shape of \( y = kx \), a straight line on a graph, where the steepness of the line is determined by the constant of variation, \( k \). This equation tells us that for every one-unit increase in \( x \), \( y \) will increase by \( k \) units. By substituting values into these equations, we can solve for unknown variables, which is exactly the skill put to the test in solving the exercise.

Solving for Variables

Solving for a variable means finding the specific value that variable should have in order to make an equation true. To solve for a variable, we often have to perform a series of algebraic manipulations. These can include addition, subtraction, multiplication, division, or even more complex operations like factoring or using the quadratic formula.

In the context of the exercise, solving for the variable \( k \) was accomplished by substituting the given coordinates into the direct variation equation and isolating \( k \) onto one side of the equation. By dividing both sides of the equation by -1, which is a simple but powerful algebraic operation, we achieved \( k = -7 \). This process of solving for a variable is a key skill in algebra because it enables us to understand and describe the relationships between different quantities.