Problem 61 - Intermediate Algebra

Question

Write a problem that can be solved by solving the system: $$ \left\\{\begin{array}{l} x+y=36 \\ \$ 1.29 x+\$ 2.29 y=572.44 \end{array}\right. $$

Step-by-Step Solution

Answer

The problem involves buying 36 fruits, with apples costing $1.29 each and oranges $2.29 each, for a total of $572.44.

Step 1: Understanding the Problem

To write a problem that this system of equations can solve, we need to identify what types of quantities the variables represent. The equation \( x + y = 36 \) suggests a total sum of items, while the second equation involves pricing, which indicates a financial context. Therefore, the problem likely involves purchasing two different types of items at different prices, with a total count and total spending given.

Step 2: Defining the Variables

Let's define \( x \) as the number of the first item and \( y \) as the number of the second item. The problem could involve purchasing two types of fruits, such as apples and oranges. The first equation \( x + y = 36 \) indicates there is a total of 36 fruits purchased, while the second equation \( 1.29x + 2.29y = 572.44 \) represents the total cost, \(572.44, with apples costing \)1.29 each and oranges $2.29 each.

Step 3: Writing the Problem Statement

Formulate a word problem based on the variables and their descriptions. For example: A person goes to a store and buys a total of 36 fruits which are either apples or oranges. Apples cost $1.29 each and oranges cost $2.29 each. If the total cost of the fruits is $572.44, how many apples and how many oranges were purchased?

Key Concepts

Variables in Word Problems

Understanding variables in word problems is crucial for solving many mathematical problems. In this exercise, variables like \( x \) and \( y \) represent unknown quantities that we need to determine. Here, \( x \) stands for the number of apples, and \( y \) indicates the number of oranges. These variables allow us to express real-life scenarios mathematically. This situation involves buying fruits where the total number of fruits and the total cost should match certain conditions.
When approaching a word problem:

• Identify what quantities the variables represent.
• Look for keywords indicating mathematical operations, such as "total" for addition.
• Express the relationships described in the problem using equations.

Clearly defining your variables is the first step in formulating a strategy to solve the word problem and find the solution.

Linear Equations

Linear equations are vital for representing relationships between variables that are proportional. In the system \( x + y = 36 \) and \( 1.29x + 2.29y = 572.44 \), we see two linear relationships.
These equations can be plotted on a graph as straight lines, hence the term "linear."
Understanding their characteristics:

• Single-variable linear equations describe lines in a one-dimensional context.
• Two-variable linear equations describe a plane intersecting in two-dimensional space.

The coherence between these equations allows for finding exact values for \( x \) and \( y \) by exploiting their intersection point on a graph, which represents the solution to the system. Linear equations are straightforward to solve using algebraic methods, providing a solid basis for approaching more complex problems.

Substitution and Elimination Methods

The substitution and elimination methods are two essential techniques for solving systems of equations like the one in the exercise, \( x + y = 36 \) and \( 1.29x + 2.29y = 572.44 \).
**Substitution Method:**

• Begin by solving one of the equations for one variable. For instance, \( x = 36 - y \).
• Substitute this expression into the other equation, replacing the variable.
• Solve the resulting single-variable equation for the unknown variable.

This method is particularly useful when one equation is easily solved for one variable.
**Elimination Method:**

• Multiply each equation by a suitable number so that one of the variables will cancel out when added or subtracted.
• Add or subtract the equations to eliminate one variable.
• Solve the resulting equation for the remaining variable.

Both methods are powerful and flexible approaches, helping you solve systems efficiently and accurately. For beginners, choosing the simpler equation for initial substitution often eases the process.