91Ó°ÊÓ

Algebraic expressions are a key component in solving linear equations. They involve using variables to represent unknown quantities, which can then be manipulated to find solutions. In the given exercise, the variables were used to represent the number of bills of different denominations.

An expression like \( 100x + 50(x + 46) + 20(7x) \) is an example of an algebraic expression. Here, each term is made up of a coefficient and a variable. In this case, the variable \( x \) represents the number of \(\$100\) bills. The coefficients (100, 50, and 20) represent the monetary value of each bill.

Algebraic expressions allow us to form equations that can be solved to find unknown values. The complexity of the expression can vary, but the process of simplifying and solving remains crucial to problem-solving in mathematics.

Problem solving in mathematics often involves breaking down a complex problem into smaller, manageable parts. This systematic approach can make a seemingly overwhelming problem easier to solve. In this exercise, the strategic steps made it possible to determine the quantities of each type of bill.

Here's a breakdown of an effective problem-solving approach:

• Understand the problem: Identify and define what you need to find. In our case, the number of each type of bill.
• Develop a plan: Use the given information to form equations. For example, creating an equation from the total value of the bills was essential.
• Execute the plan: Simplify and solve the equations step by step, as demonstrated through variable manipulation and equation solving.
• Verify the solution: Check the final values by calculating the total value to ensure it aligns with the original problem conditions.

Using this method ensures clarity and helps in arriving at an accurate solution.

Variable manipulation is at the heart of solving linear equations. It involves performing operations to isolate the variable in question. In this exercise, the goal was to determine \( x \), the variable representing the number of \(\$100\) bills.

The initial step was setting up the equation based on the given conditions: \( 100x + 50(x + 46) + 20(7x) = 9550 \). By expanding and combining like terms, the equation becomes simpler and more manageable. Operations like addition, subtraction, multiplication, and division are employed to "manipulate" the equation, gradually isolating \( x \).

• Expanding: Distribute coefficients to terms. Example: \( 50(x + 46) \) becomes \( 50x + 2300 \).
• Combining like terms: Group similar terms to simplify. Example: \( 100x + 50x + 140x \) simplifies to \( 290x \).
• Isolating the variable: Use inverse operations to solve for \( x \). Subtract \( 2300 \) from both sides, then divide by \( 290 \) to solve for \( x \).

This methodical approach is fundamental to algebra and helpful in solving various real-world problems efficiently.