Lesson 2.6

Modeling with Two-Steps

Real life rarely hands you an equation on a silver platter. It hands you a messy paragraph. Your job is to translate that story into math.

Introduction

Most real-world situations follow a simple pattern: a starting amount plus a rate of change equals a total.

Past Knowledge

You can translate phrases like "5 more than twice a number" into .

Today's Goal

Write and solve two-step equations from word problems.

Future Success

This is the foundation for Linear Functions () coming in Unit 5.

Key Concepts

The "Flat Fee + Rate" Pattern

Variable Amount + Flat Fee = Total

Variable Amount

("per", "each", "every")

Flat Fee

+ b

("start", "fee", "initial")

Total

= y

("is", "equals", "total")

Keyword Decoder

Rate Look for: per, each, every, rate, monthly.
Start Look for: flat fee, starting text, deposit, one-time cost, discount (-).
Total Look for: equals, is, total cost, final amount.

Worked Examples

Example 1: The Shopping Trip

Basic

Liam bought a magazine for $5 and 4 erasers. He spent a total of $13. How much did each eraser cost?

Define Variables

Let = cost of one eraser.

4 erasers = .

Write Equation

Solve

Divide by 4:

Answer: Each eraser cost $2.

Example 2: The Negative Rate

Intermediate

A 350-gallon pool is draining at a rate of 20 gallons per minute. How long until it considers 150 gallons remaining?

Identify Key Info

Start: 350 (Positive)
Rate: 20 gallons/min (Negative, because draining)
Total: 150 (Remaining)

Write Equation

Solve

Divide by -20:

Answer: 10 minutes.

Common Pitfalls

Confusing the Rate and Start

"20 per week" means 20 is the rate (). "Starts with 20" means 20 is the constant ($+20$). Look for the word "per"!

Handling Discounts

Words like "discount," "fewer," or "drained" mean subtraction. If you use addition, your answer will be wrong.

Real-Life Applications

Freelancers use this every day to quote jobs. "I charge a $200 deposit plus $50 per hour."

If a client has a budget of $500, the freelancer can easily calculate how many hours they can afford to work.