Problem 45 - Algebra 1: Concepts and Skills

Question

The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates x and y .\( \text { (Lesson } 4.6)\) $$ x=3, y=-6 $$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) when they vary directly is \(y=-2x\).

Step 1: Find the constant of proportionality (k)

To find the constant of proportionality, rearrange the equation \(y=kx\) to solve for \(k\). The new equation becomes \(k=y/x\). Substitute \(x=3\) and \(y=-6\) into equation to calculate \(k\). So \(k=(-6)/3=-2\).

Step 2: Write the equation

Substitute the obtained \(k=-2\) into the equation \(y=kx\). The final equation that relates \(x\) and \(y\), when they vary directly, is \(y=-2x\).

Key Concepts

Constant of Proportionality

Understanding the constant of proportionality is crucial when studying direct variation. In essence, this constant, often denoted as 'k', is the factor that relates two quantities varying directly with each other. If you increase one quantity, the other increases by a proportionate amount. The direct variation formula, \( y = kx \), embodies this concept, where 'y' and 'x' are the two variables in direct variation, and 'k' is their constant of proportionality.

To find 'k', we divide the one variable by the other, as the solution to the exercise illustrates. If you have \( x = 3 \) and \( y = -6 \) and you know they vary directly, you simply divide \( y \) by \( x \) to get \( k = -6/3 = -2 \). Here, '-2' is our constant of proportionality, indicating that for every unit increase in 'x', 'y' decreases by 2 units, highlighting that 'k' can be negative as well, illustrating a decrease rather than an increase.

Proportional Relationships

Proportional relationships describe the scenario where two quantities increase or decrease at the same rate. In a direct variation, which is a type of proportional relationship, this uniform change is captured by the constant of proportionality. Proportional relationships have some telltale characteristics worth noting:

• They pass through the origin (0,0) on a coordinate plane as neither variable exists without the other in direct variation.
• The ratio of the variables is constant (hence, 'constant of proportionality').
• On a graph, they're represented by a straight line.

Through proportional relationships, if you know one variable, you can always find the other. If \( x \) is tripled in the problem, for example, \( y \) would be tripled too, only in the opposite direction (positive to negative or vice versa) if 'k' is negative.

Algebraic Equations

Algebraic equations serve as the backbone of interpreting mathematical relationships. In the context of direct variation, the algebraic equation \( y = kx \) succinctly illustrates how two variables are connected. This equation allows us to calculate unknowns and predict outcomes based on established relationships.

Whenever faced with word problems or data that implies a constant rate of change between two variables, forming an algebraic equation can simplify the analysis. From a given pair of variables, as in the exercise (\( x=3, y=-6 \) ), we constructed the equation \( y = -2x \) by determining the constant of proportionality. This equation can now be used to predict the value of 'y' for any 'x' and vice versa, showcasing the power of algebra in solving real-world problems.