Problem 64 - Intermediate Algebra

Question

A woman paid \(\$ 219\) for two blouses and four pairs of pants. If we let \(x=\) the cost of a blouse and \(y=\) the cost of a pair of pants, an equation modeling the purchase is \(2 x+4 y=219\) Explain why there is not enough information to determine the cost of a blouse or the cost of a pair of pants.

Step-by-Step Solution

Answer

There isn't enough information because one equation with two variables suggests infinite solutions.

Step 1: Understand the given equation

The equation given is \(2x + 4y = 219\), where \(x\) is the cost of a blouse and \(y\) is the cost of a pair of pants. Here, \(2x\) represents the total cost of two blouses and \(4y\) represents the total cost of four pairs of pants.

Step 2: Identify the type of problem

This problem involves a single equation with two unknowns, \(x\) and \(y\). To uniquely determine both variables, we generally need as many independent equations as we have variables.

Step 3: Analyze the information provided

The information provided gives us a single equation: \(2x + 4y = 219\). With only one equation, any number of \(x\) and \(y\) pairs can satisfy it, leading to an infinite number of solutions.

Step 4: Consider what is missing

In general, to find unique values for two variables, you need two independent equations. Here, we are missing another independent piece of information, such as the cost of either a blouse or a pair of pants, or another equation involving \(x\) and \(y\).

Key Concepts

Systems of Equations

To understand the woman's purchase, we first encounter the concept of a system of equations. A system of equations is a set of multiple equations that are all true simultaneously. In problems involving systems of equations, we aim to find the values of variables that satisfy all the equations in the system concurrently. In this exercise, we have a potential system involving the cost of blouses and pants.When dealing with such problems, it's like solving a puzzle: you use different pieces of information (equations) to figure out the unknowns. A single equation like the one here, \(2x + 4y = 219\), is not enough to solve for two variables \(x\) and \(y\). To find unique values for each item (cost of a blouse and a pair of pants), we need an additional independent equation containing these variables.This is akin to having one piece of a puzzle when you need two interconnected ones to see the complete picture. Without the second equation, various combinations of \(x\) and \(y\) can work, creating what's called an infinite set of solutions.

Variables and Unknowns

In our exercise, understanding variables and unknowns is essential. A variable is a symbol that represents a number in equations and problem scenarios; it's an unknown part of the story. Here, \(x\) and \(y\) are our variables. The value of these variables is unknown to us at the start - \(x\) being the cost of a blouse, and \(y\) the cost of a pair of pants.It's crucial to identify your variables correctly to set up your equation, as that's the starting point of solving. Each variable can stand for different numerical values, and our goal is to determine their exact value in a given context or equation. In mathematical terms, they provide flexibility and a way to generalize patterns and relationships, which is why they are often called 'unknowns.' In this exercise, since we have only one equation but two unknowns, we cannot uniquely determine both variables without additional information.

Solving Equations

Solving equations means finding the value of the unknowns that makes the equation true. In this exercise, we are given the equation \(2x + 4y = 219\). Our task is to determine the values of \(x\) and \(y\). However, solving for two unknowns requires at least two independent equations.Here's why: each equation represents a condition that the variables must satisfy. With only one equation and two variables, there are infinitely many solutions because the values of \(x\) and \(y\) can vary in many ways and still satisfy the equation. For example, if someone also said the cost of one blouse is \(\$30\), then we could substitute \(x = 30\) into \(2x + 4y = 219\) and solve the resulting equation for \(y\). This would provide a specific solution for both \(x\) and \(y\).Always remember, the number of equations needed to find a unique solution must match the number of variables. Thus, to solve equations involving multiple variables precisely, seek additional independent information or equations.