Solving Equations: One Step at a Time

Lesson 3 of 4: Expressions and Equations

In this lesson:

• Solve equations using inverse operations
• Use the 5-step protocol to write and solve word problems

What You Will Learn Today

1. Classify equations as additive () or multiplicative ()
2. Solve both types using inverse operations
3. Verify solutions by substitution
4. Write equations from word problems using the 5-step protocol
5. Interpret solutions in context

Emma's Money — Where Do We Start?

Emma had some money. She spent $12.50 and has $7.25 left.
How much did she start with?

• You could almost answer this with just arithmetic
• But which arithmetic? Add? Subtract? From which number?
• The equation tells you exactly what to do — and why

The Balance Model: Equality Stays Level

The scale tips if you change only one side

Two Types, Two Inverse Operations

Before you compute — ask:

1. Which type?
2. Which inverse?

Worked Example: Solving an Additive Equation

Type: Additive — 3 is added to
Inverse: Subtract 3 from both sides

Check: Does ? ✓

Decimal Extension: Same Steps, New Numbers

Solve:

Type: Additive — 6.5 is added to
Inverse: Subtract 6.5 from both sides

Check: Does ? ✓

Solving the Multiplicative Type:

Type: Multiplicative — is multiplied by 4
Inverse: Divide both sides by 4

Check: Does ? ✓

Fractional Coefficient: Multiply by the Reciprocal

The coefficient is .
Dividing by = multiplying by

Check: ✓
Watch out: — that's dividing the wrong way!

Name the Type Before Computing

For each equation, name the type and the inverse — before computing:

Write "additive/subtract" or "multiplicative/divide" for each.

Practice: Solve and Check Both

Solve each equation and verify your solution:

Show the inverse step and the check for each.

Building the Equation from Context

You now know how to solve an equation — once you have it.

But where does the equation come from?

Real situations don't arrive labeled "."
You read the context, identify the unknown, and build the equation.

That's what the 5-step protocol teaches next.

The Five-Step Word Problem Protocol

Write every step label on your paper — they enforce the discipline.

Applying All Five Steps: Maria's Earnings

From last lesson: .

Step 2: Let = the number of hours Maria worked
Step 3:
Step 4: . Check: ✓
Step 5: Maria worked 5 hours.

Bookstore Problem: Reading for Additive Structure

After selling 34 books, 51 remain. Original shipment?

Step 2: Let = books in the original shipment
Step 3:
Step 4: . Check: ✓
Step 5: The original shipment had 85 books.

Quick Check: Problem A's Structure

For :

• Is this additive or multiplicative?
• What inverse operation did we use — and why?
• Would the check catch an error if we had added wrong?

Answer each question before the next slide.

Recipe Problem: Fractional Coefficient in Action

Each batch uses cup sugar. You have 9 cups. Batches?

Step 2: Let = the number of batches
Step 3:
Step 4: . Check: ✓
Step 5: You can make 12 full batches.

Spot the Error: Emma's Starting Amount

Emma spent $12.50 and has $7.25 left. Which setup is correct?

• Wrong A:
• Wrong B:
• Correct:

Emma's Problem: Solving the Right Way

Step 2: Let = the amount Emma started with
Step 3:
Step 4: . Check: ✓
Step 5: Emma started with $19.75.

Guided Practice: Plumber Charges $45/hr

A job cost $202.50. How many hours?

Step 2: Let = ___________________________
Step 3: Write the equation: _______ = _______
Step 4: Solve — what type? What inverse?
Step 5: Write your interpretation sentence.

Complete each step before continuing.

Apply All Five Steps: Your Turn

"A cyclist rode a total of 84 miles over several days. She rode the same distance each day and finished in 6 days. How far did she ride each day?"

Apply all five steps — no scaffolding this time.

Write each step label, then complete it.

Three Mistakes Students Make Most Often

Wrong inverse: → . Subtract 5 from both sides.

Inverted division: → . Coefficient divides into : .

Backward setup: Write for situation, not text order. Ask: unknown? happened? remains?

What You Can Now Do

✓ Classify and solve both equation types
✓ Isolate with the matching inverse
✓ Check by substitution
✓ Write equations from context; interpret answers

Classify first — wrong type → wrong inverse
Fractional coefficient: use the reciprocal
Backward setup: role, not text order

Next: Inequalities Extend This Thinking

• One equation → one solution
• One inequality → infinitely many solutions
• The balance model still works — with one new rule to learn
• 6.EE.B.8: equals sign becomes an inequality sign