Problem 29 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=-1, y=-1$$

Step-by-Step Solution

Answer

The final equation that relates x and y is y = x.

Step 1: Identify the given values

The given values are x = -1 and y = -1.

Step 2: Substitute the given values into the direct variation formula

Substitute x = -1 and y = -1 into the formula y = kx. It will become -1 = k*(-1).

Step 3: Solve for k

Solving the equation for k gives k = 1.

Step 4: Write the equation that relates x and y

Substitute k = 1 into the formula y = kx to get the equation y = x. This is the final equation that relates x and y.

Key Concepts

Algebraic Equations

In the realm of mathematics, algebraic equations are the cornerstone of understanding how different quantities relate to each other. They consist of symbols and numbers arranged in a formula that represents a relationship where both sides are equal. For instance, when we have variables like x and y that vary directly, an algebraic equation can describe this relationship succinctly.

In our context, when x and y are given as -1, we can establish an algebraic equation by using these values. In solving for the equation y = kx, where k represents a constant proportionality, substituting the values of x and y allows us to unravel the mystery of this constant. The beauty of algebraic equations lies in their ability to be re-arranged and solved for any variable, providing a powerful tool for uncovering relationships between variables.

Proportionality Constant

The proportionality constant, denoted as k in our example, is the key to understanding direct variation. It is the unchanging number that relates two variables that are directly proportional to each other. In simpler terms, think of it as a multiplier that scales one variable with respect to the other.

When two variables, such as x and y, change at the same rate, this constant ensures they remain in sync. If the value of k is positive, it indicates that both variables increase or decrease together. Conversely, if k is negative, one variable will increase while the other decreases. In our exercise, after substituting the values into y = kx and finding that k equals 1, we see that for every unit change in x, y changes by the same amount. This constant proportionality forms the essence of direct variation and provides a predictable consistency in mathematical relationships.

Variable Relationships

Understanding the relationships between variables is vital in mathematics. Variables can relate to each other in various ways; they might be directly proportional, inversely proportional, or have no relationship at all.

In the case of direct variation, as one variable changes, the other changes in a consistent manner, as set by the proportionality constant k. This constant serves as the anchor of their relationship. If k is positive, the variables increase and decrease in unison, whereas a negative k means they move in opposite directions. Our exercise illustrates a direct proportionality since k is determined to be 1, showcasing a perfect one-to-one increase. This type of understanding is crucial, not just in algebra, but also in real-world applications where predicting one variable's behavior based on another can be essential, such as in physics or economics.