Solving Equations: Two Algebraic Forms

Lesson 1 of 2

In this lesson:

• Solve equations of the form
• Solve equations of the form
• Interpret solutions in context

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Write an equation of the form or for a real-world situation
2. Solve both equation forms using inverse operations
3. Check your solution by substituting back into the original equation
4. Interpret what the solution means in context

Equations as Perfectly Balanced Scales

Every operation done to one side must be done to the other — or the balance breaks.

Inverse Operations: The Key to Solving

To isolate the variable, undo each operation in reverse order:

Work in reverse order of operations — undo addition/subtraction first, then multiplication/division.

Solving Equations with Integer Coefficients

Solve:

Step 1: Subtract 3 from both sides

Step 2: Divide both sides by 2

Check: ✓

Solving Equations with Decimal Coefficients

Solve:

Step 1: Add 1.5 to both sides

Step 2: Divide both sides by 0.5

Check: ✓

Solving Equations with Fraction Coefficients

Solve:

Step 1: Subtract 2 from both sides

Step 2: Multiply both sides by (reciprocal)

Check: ✓

Quick Check: Identify the First Step

What is the first step to solve ?

A) Divide both sides by 5

B) Subtract 7 from both sides

C) Add 7 to both sides

D) Multiply both sides by 5

Think before you advance — which operation is being undone first?

Your Turn: Solve Four Equations

Solve each equation. Show your work and check each answer.

4. (solution is positive)

Pause and solve before advancing.

Answers: Check Your Four Solutions

1. → →
2. → →
3. → →
4. → →

Check each: substitute back into the original.

A Second Form:

Example 1: Divide First Is Cleaner

Solve:

Method 1 — Divide first (cleaner):

Method 2 — Distribute first:

Check: ✓

Why is Method 1 cleaner here? 24 ÷ 6 = 4 (integer).

Example 2: Distribute First Is Cleaner

Solve:

Method 1 — Divide first (messy):

Method 2 — Distribute first (cleaner):

Check: ✓

25 ÷ 3 is a repeating decimal — distributing gives cleaner arithmetic.

Which Method? A Decision Guide

Both methods always work — this flowchart just shows the faster path.

Guided Practice: Your First Step

Solve:

Step 1 is given: divide both sides by 5

Your turn: What is step 2? What is ?

Now check your answer in the original equation.

Advance when you have and have verified it.

Which Method Would You Choose?

For :

A) Divide first — because (clean)

B) Distribute first — because is messy

C) Either method works — I'd do both to verify

Pause and decide before advancing.

Your Turn: Four Equations in Second Form

Solve each. Label which method you used (divide first or distribute first).

Pause and solve. Show your method label.

Answers: Four p(x + q) = r Equations

1. → divide first →
2. → divide first →
3. → distribute first →
4. → divide first →

Misconception Check: Inside First Error

Mistake: Subtracting 3 first from gives — wrong

• means 5 multiplies the entire
• Cannot subtract 3 without removing the 5 first
• Fix: divide by 5 or distribute — both handle the 5 first

Key Takeaways: Two Equation Forms

✓ Inverse operations in reverse order — undo addition, then multiplication

✓ For : divide first if is clean, otherwise distribute

✓ Always substitute back to verify

Undo addition with subtraction, not more addition

Don't subtract inside parentheses before removing the multiplier

Preview: Deck 2 — Inequalities and Word Problems

Coming up next:

• Solving inequalities: same steps, one new rule (the flip)
• Graphing solutions on a number line
• Writing equations and inequalities from word problems
• The 4-step word problem framework

Deck 2 builds directly on the equation-solving skills you practiced today.