Problem 54 - Algebra 1

Question

In an inverse variation, the product \(x y\) is constant. If \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) are solutions of \(x y=k,\) then \(x_{1} y_{1}=x_{2} y_{2} .\) Use this equation to find the missing value. Find \(x_{2}\) when \(x_{1}=9, y_{1}=-3,\) and \(y_{2}=12\).

Step-by-Step Solution

Answer

So, the missing value \(x_{2}\) equals -2.25.

Step 1: Refer to the Property of Inverse Variation

By the property of inverse variation, we know that the product of \(x\) and \(y\) is a constant. Hence, for any pair of \(x_i\) and \(y_i\), \(x_{1}y_{1} = x_{2}y_{2}\).\n

Step 2: Plug in the Given Values

Now, substitute the given values into the equation from Step 1. We know \(x_{1} = 9\), \(y_{1} = -3\), and \(y_{2} = 12\). We just need to find \(x_{2}\). Substituting, we get \(9*-3 = x_{2}*12\).\n

Step 3: Solve for \(x_{2}\)

Now solve the equation for \(x_{2}\). So, rearrange the equation to find \(x_{2}\): \(x_{2} = (9*-3) / 12.\)\n

Key Concepts

Constant Product

In the realm of algebra, the term 'constant product' refers to the unchanging result of multiplying two variables when they are in an inverse variation relationship. This means, if one variable increases, the other decreases at a rate that keeps their product the same. For example, if we denote two variables as \(x\) and \(y\), and their constant product as \(k\), we can express this relationship algebraically as \(xy = k\).

In our given exercise, the task was to demonstrate this concept by finding a missing value in a set of inverse variations, where \(x_1y_1 = x_2y_2\). When we're given three out of the four variables, the constant product allows us to solve for the unknown with ease, showcasing the powerful utility of this mathematical property.

Solving Inverse Variation Problems

To solve inverse variation problems, we utilize the principle that \(x\) and \(y\) maintain a constant product. This property allows us to create an equation based on known values that can be manipulated to find the missing ones. It's a game of balance where adjusting one value compensates for the change in the other.

Step-by-Step Approach

• Establish the inverse variation equation \(xy = k\), where \(k\) is the constant product.
• Insert the known values into the equation.
• Rearrange the equation to isolate and solve for the unknown variable.

This systematic approach was used to determine \(x_2\) in the given exercise by substituting the known values into the inverse variation formula and isolating \(x_2\) to find its value.

Proportional Relationships

While inverse variation deals with a constant product, proportional relationships are about maintaining a constant ratio. A direct proportion means that as one variable increases, the other does too, at a rate that keeps the ratio the same, expressed as \(y = kx\), where \(k\) is the constant ratio.

Understanding these different relationships is critical, as they are foundational concepts in algebra that describe how variables interact with each other. Recognizing whether a relationship is inverse or direct is crucial when solving various mathematical problems and can be particularly advantageous in fields such as physics and economics, where such proportional dynamics frequently occur.

Algebraic Equations

Algebraic equations are the backbone of solving mathematical problems involving unknowns. They allow us to formalize relationships between variables and constants, dissecting complex scenarios into solvable expressions. When we encounter an inverse variation, the algebraic equation elegantly captures this relationship and gives us the tools to unlock missing pieces of the puzzle.

An equation is like a balance scale; whatever you do to one side, you must do to the other to maintain equilibrium. This balancing act enables us to manipulate equations to isolate and solve for unknowns, as seen in the inverse variation problem from our exercise.