Welcome back to class! Today, we tackled one of the most important (and sometimes most intimidated) topics in Algebra: Systems Word Problems. If you've ever wondered, "When will I ever use this?" today is the answer. We are taking real-world scenarios—like calculating money in your pocket or comparing gym memberships—and using math to find the solution.

The Golden Rule: Define Your Variables

Before writing a single number, you must decide what you are looking for. As we discussed in the class notes, you always start by defining your variables. Usually, these are the two things the problem asks you to find.

Case Study 1: The Coin Problem

In class, we looked at a classic problem: Maribel has $1.25 in her pocket in quarters and dimes. She has a total of 8 coins. How many of each does she have?

Step 1: Define Variables
Let $q =$ number of quarters
Let $d =$ number of dimes

Step 2: Create the System
We need two equations. One usually deals with the quantity (how many physical items), and the other deals with the value (how much they are worth).

  • Quantity Equation: $q + d = 8$ (Total coins)
  • Value Equation: $0.25q + 0.10d = 1.25$ (Total value)

Step 3: Solve
In the attached handwritten notes, we used Elimination to solve this. By manipulating the equations, we found that $d = 5$. Plugging that back in ($q + 5 = 8$), we found that $q = 3$.

Answer: Maribel has 3 quarters and 5 dimes.

Case Study 2: Comparing Costs (The Gym Problem)

Another common scenario is comparing two services to see when they cost the same. We looked at "Fabulously Fit" vs. "The Fitness Studio."

  • Fabulously Fit: $35 per month + $50 enrollment fee.
  • The Fitness Studio: $40 per month + $35 enrollment fee.

We set up our system using $C$ for Cost and $m$ for Months:

  • Equation 1: $C = 35m + 50$
  • Equation 2: $C = 40m + 35$

Since both equations equal $C$, we used the Substitution Method (setting them equal to each other):

$$35m + 50 = 40m + 35$$

After solving for $m$, we discovered that at 3 months, the cost is the same ($155). This is the "break-even" point!

Comparing the Three Methods

We also reviewed the "Three Methods Worksheet." Remember, whether you use Graphing, Substitution, or Elimination, you should arrive at the same solution (the point of intersection).

  • Graphing: Good for visualizing the solution, but can be messy if the answer involves fractions (like Example 4 in the PDF where the answer was $(1, -7/2)$).
  • Substitution: Best when a variable is already isolated (like $y = ...$ or $C = ...$).
  • Elimination: Best when both equations are in Standard Form ($Ax + By = C$).

Homework

Please complete the remainder of the worksheet attached below. Practice setting up the equations first—that is the hardest part! Once the equations are set, use the method that feels most comfortable to you.