Equations Wearing a Costume

Fractions, Decimals, and Multi-Step Solving

In this lesson:

• Clear fractions and decimals to simplify equations
• Distribute, collect, and solve multi-step equations

A Recipe Hands You a Fraction

A batch needs cup of flour. You have cups to spend.

How many batches can you make — and how do you solve it cleanly?

What You Will Learn To Do Today

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

1. Solve equations with fraction coefficients, including by clearing fractions
2. Solve equations with decimal coefficients by clearing decimals
3. Distribute and collect like terms to solve an equation
4. Solve multi-step equations combining all of these skills
5. Verify a solution by substituting it back in

Two Ways to Solve It

• Direct: subtract , then multiply by →
• Clear fractions: multiply every term by → →

Multiplying both sides by 3 is the multiplication property of equality.

Clearing Fractions With the Common Denominator

To clear fractions, multiply every term by the LCD.

• The fractions all turn into whole numbers
• You're left with an equation you already know how to solve
• "Multiply both sides" means multiply every term

When Clearing Fractions Really Shines

The LCD of , , and is . Multiply every term:

Check Every Term Got Multiplied

In , multiplying by :

The middle term is the one people forget.

Decimals Are the Same Idea

• Direct: subtract , divide by →
• Clear decimals: multiply every term by →

Since , multiplying by clears it — just like an LCD.

Watch the Direction of the Division

Solving , isolate :

"Zero-point-three of is 2.7" — so is bigger than 2.7, not tiny.

Your Turn: Pick a Strategy

• Notice
• Clear by multiplying by , or by
• Either works — which is cleaner here?

New Tools for Messy Structure

Clearing fractions handled ugly numbers. Now: ugly structure.

• Distribute to remove parentheses
• Collect like terms to simplify each side
• Then solve the equation you already know

Distribute to Every Term Inside Parentheses

• Distribute:
• Note , not
• Collect left:

Your Turn: Finish Solving This One

From :

• Subtract from both sides
• Add to both sides
• Then divide — what is ?

Work each step, then advance to check.

Say "Subtract," Not "Move It"

Say it as an operation on both sides:

The sign change is a result of subtracting — not a separate rule.

Collect With Variables on Both Sides

• Combine left:
• Subtract , add :
• Divide by :

Combine Every Skill in One Equation

The Same Equation, Distributing First

• Distribute:
• Still solvable — but every step carries fractions

Both paths give . Which would you choose?

Find the Error in This Expansion

A student expands as:

What did they forget? Write the correct expansion.

Quick Check: Solve This Solo

• Solve it on your own
• Then substitute your answer back in

Commit to an answer before you advance.

Multi-Step, From Start to Finish

Solve completely, then check:

No scaffolding. Clear fractions, distribute, collect, solve, verify.

When the Answer Is "No Solution"

• Clear and simplify carefully
• The variable cancels and leaves a false statement
• No solution — exactly the case from 8.EE.C.7.a

Four Common Traps to Avoid Here

Distribute to every term — , not
When clearing, multiply every term, even whole numbers
"Move it" really means subtract from both sides — keep the sign right
To undo , multiply by ; don't flip the division

What You Now Understand About Solving

✓ Clearing fractions or decimals turns hard equations into familiar ones
✓ Distribute and collect to tame any structure
✓ A messy equation is a clean one in a costume

Coming Up Next: Solving Systems

You can now solve any single linear equation.

Next you'll solve systems — two equations at once (8.EE.C.8). The clearing, distributing, and collecting you mastered today are the engine inside substitution and elimination.