Solving Equations

By the end of this lesson you will be able to:

• Solve one-step equations using the balance method
• Solve two-step equations by undoing in reverse order
• Handle equations with the unknown on both sides and brackets
• Form and solve equations from word problems

I'm Thinking of a Number…

"If I add 7 to my number, I get 12. What's my number?"

You might guess 5 straight away — and you'd be right.

But what if the unknown appears on both sides? Guessing stops. The balance method always works.

Recall: What Undoes Each Operation?

Every operation has an inverse that undoes it.

Quick check: What undoes ? What undoes ?

An Equation Has Two Sides That Balance

An equation has an sign — both sides are equal, like a balanced scale.

An expression like has no sign. An equation like has exactly one value of that makes it true.

Balance Method: Solving

Subtract 7 from both sides:

Check: ✓

Also: → divide both by 3 → . Check: ✓

Find the Error: One-Side Mistake

Student wrote:

Check: ✗ — wrong.

Error: operated on one side only.

Fix: subtract 7 from both sides → . Check ✓

Solving Three Types of One-Step Equations

• . Check: ✓
• . Check: ✓
• . Check: ✓

Check Your Work: One-Step Equations

Solve each equation and substitute to check.

Answers: ; ;

Two Traps: Which Operation First?

"I multiply a number by 2, then add 3. I get 11."

In algebra:

Two operations on . To undo both — reverse first the or the ?

(Think before the next slide.)

Reverse Order: Constant First, Then Coefficient

Rule: Undo or before or

Check: ✓

Guard: Dividing first gives — fractions appear unnecessarily.

Two-Step Equations: Two Worked Examples

. Check: ✓

. Check: ✓

Constant first, coefficient second — every time.

Check Your Work: Two-Step Equations

Solve each equation and check by substitution.

(Which inverse first, and why?)

Answers: ;

The Unknown on Both Sides

"5 of my number equals 2 of my number plus 9."

appears on both sides. We can't isolate it while a copy sits on the right.

Move: subtract from both sides to collect all -terms on the left.

Collecting Variable Terms to One Side

— subtract from both sides:

Check: ✓

Guard: Sign Error When Collecting

Student wrote:

Error: added instead of subtracting. Check fails: — not a solution.

Fix: subtract from both sides → ✓

Brackets First:

Step 1: Expand —

Step 2: Strip constant —

Step 3: Divide —

Check: ✓

Expand before moving any terms — the bracket hides the .

Check-In: Both Sides and Brackets

Solve and verify by substitution.

(Q1: collect first. Q2: expand first.)

Answers: ;

From Solving to Forming an Equation

You can solve any of these:

• One-step equations
• Two-step equations
• Unknown on both sides; brackets

The last skill: turning a word problem into an equation. Once you form it, solving is something you already know how to do.

Five Steps for Every Word Problem

1. Name — let = the unknown
2. Form — write the equation
3. Solve — use inverse operations
4. Verify — substitute back
5. State — answer in words

Worked Word Problem: "Think of a Number"

Problem: Think of a number. Multiply by 3 and add 4. The result is 19.

1. Let = the number
4. Check: ✓
5. The number is 5.

Thread Closer: The Perimeter Problem

Perimeter 20 cm, width 4 cm — find the length .

Check: ✓ The length is 6 cm.

Your Turn: Full Five-Step Routine

A trader sells 3 bags of maize flour at the same price, then earns 500 UGX extra. Total: 8,000 UGX. What is the price per bag?

Apply all five steps. No scaffolding here.

Answer: 2,500 UGX per bag

An Equation Is a Balance

Whatever you do to one side, do to the other.

• Strip or before or
• Collect variables: subtract from both sides
• Expand brackets before moving terms
• Substitute back to verify

Next: algebra-04 — same method, one new rule.