Problem 25 - Algebra 1: Concepts and Skills

Question

The variables \(x\) and \(y\) vary inversely. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=45, y=0.6 $$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) is \(x = 27/y\).

Step 1: Understanding the inverse relation

The variables \(x\) and \(y\) are said to vary inversely, which means that their relationship can be represented by an equation of the form \(x = k/y\), where \(k\) is the constant of variation.

Step 2: Substitute the known values of \(x\), \(y\) into the equation

Subsitute \(x = 45\) and \(y = 0.6\) into the equation to find the value for \(k\). Using these values, the equation becomes: 45 = \(k / 0.6\).

Step 3: Solve for \(k\)

Multiply both sides by 0.6 to solve for \(k\), which yields \(k = 27\)

Step 4: Write the equation

Substitute \(k = 27\) back into the equation, the relation between \(x\) and \(y\) is written as \(x = 27/y \).

Key Concepts

Inverse Relationship

An inverse relationship is a fundamental concept in algebra where two variables, let's call them x and y, are connected in such a way that when one of them increases, the other decreases. This kind of relationship can seem a bit counter-intuitive at first, but it's actually quite prevalent in our world. Think of it like a seesaw: when one side goes up, the other goes down.

Mathematically, when we say that x and y vary inversely, we mean that their product is constant. This can be represented by the equation \(xy = k\text{, where }k\) is a non-zero constant. In other words, as x gets larger, y must get smaller in such a way that their product always equals k. For instance, if you double the value of x, y will have to halve to maintain this constant product.

Constant of Variation

The constant of variation, usually denoted by k, is the 'glue' that maintains the inverse relationship between two variables. It doesn't change, hence the term 'constant'. Identifying this number is crucial as it defines the specific relationship between the variables in your equation.

Using the seesaw analogy, k would be like the pivot point—it stays put whilst the seesaw moves up and down. It's the fixed value that tells us the ratio or product of the variables involved. In our exercise example <\(x = k/y\)> where x and y vary inversely, the constant of variation k stays the same no matter what values x and y take, as long as their product equals k.

Solving for Constants

Whenever you come across problems involving inverse variation, 'solving for constants' means finding the value of the constant of variation k. This step is vital in understanding the specific relationship between your variables and is the key to creating an equation that represents their relationship.

To solve for k, simply plug in the values provided for x and y into the inverse variation formula <\text{\(x = k/y\)\}}. In our textbook example, substituting x = 45 and y = 0.6 into the equation allows us to calculate the constant k. Often this involves some basic algebra-manipulating steps; multiplying both sides by y, to isolate k on one side of the equation. The ease of this process will depend on your familiarity with algebraic operations and confidence in rearranging equations.