LESSON 18 / 100 | 18% COMPLETE | STAGE II — EQUATIONS
Stage II — Equations
One-Step Equations (× and ÷)
Division undoes multiplication. Multiplication undoes division. One step — variable free.
Section 01
The Two Forms
ax = b
Inverse: divide both sides by a
x = b / a (a ≠ 0)
x / a = b
Inverse: multiply both sides by a
x = b × a
If multiplication packages the variable — multiplying it by a factor — then division unwraps it. If division has hidden the variable inside a fraction, multiplication restores it to the open. The goal in both cases is the same: x alone on one side, a number on the other.
Section 02
Solving — Multiplication Equations
Worked Example A — Solve 5x = 35
5x
=
35
5x / 5
=
35 / 5
divide both sides by 5
x
=
7
Verify
LHS: 5(7) = 35 = RHS ✓
Worked Example B — Solve −4x = 28
−4x
=
28
−4x / (−4)
=
28 / (−4)
divide both sides by −4
x
=
−7
positive ÷ negative = negative
Verify
LHS: −4(−7) = 28 = RHS ✓
Worked Example C — Solve −6x = −42
−6x / (−6)
=
−42 / (−6)
divide by −6
x
=
7
negative ÷ negative = positive
Verify
LHS: −6(7) = −42 = RHS ✓
Section 03
Solving — Division Equations
Worked Example D — Solve x / 6 = 9
x / 6
=
9
(x / 6) × 6
=
9 × 6
multiply both sides by 6
x
=
54
Verify
LHS: 54 / 6 = 9 = RHS ✓
Worked Example E — Solve x / (−3) = 7
(x / −3) × (−3)
=
7 × (−3)
multiply both sides by −3
x
=
−21
Verify
LHS: (−21) / (−3) = 7 = RHS ✓
Section 04
Equations with Fraction Coefficients
When the coefficient of x is a fraction, multiplying both sides by its reciprocal isolates x in one step. This is more efficient than dividing by the fraction.
Reciprocal Method
To solve (a/b)x = c, multiply both sides by b/a (the reciprocal of a/b):
x = c × (b/a) = bc/a
Worked Example F — Solve (3/4)x = 12
(3/4)x
=
12
(4/3) × (3/4)x
=
12 × (4/3)
multiply by reciprocal 4/3
x
=
48/3 = 16
Verify
LHS: (3/4)(16) = 48/4 = 12 = RHS ✓
Worked Example G — Solve (−2/5)x = 8
(−5/2) × (−2/5)x
=
8 × (−5/2)
multiply by reciprocal −5/2
x
=
−20
8 × (−5/2) = −40/2 = −20
Verify
LHS: (−2/5)(−20) = 40/5 = 8 = RHS ✓
Section 05
Sign Rules When Dividing
| Equation | Coefficient sign | RHS sign | Solution sign |
|---|---|---|---|
| 4x = 20 | positive | positive | positive (+5) |
| 4x = −20 | positive | negative | negative (−5) |
| −4x = 20 | negative | positive | negative (−5) |
| −4x = −20 | negative | negative | positive (+5) |
Critical Habit — Never Drop the Negative Coefficient
When dividing by a negative coefficient, both the coefficient and the RHS must be divided. The most common error is dividing only the RHS by the absolute value of the coefficient and losing the sign interaction entirely.
✗ Wrong: −4x = 20 → x = 20/4 = 5
✓ Correct: −4x = 20 → x = 20/(−4) = −5
Section 06
Word Problems
Word Problem A — Six friends share a restaurant bill equally. Each pays R87. What was the total bill?
1
Let b = total bill in rands
2
Equation: b / 6 = 87
3
Multiply both sides by 6: b = 87 × 6 = 522
4
The total bill was R522. Verify: 522 ÷ 6 = 87 ✓
Word Problem B — A number multiplied by −7 gives 56. Find the number.
1
Equation: −7n = 56
2
Divide by −7: n = 56 / (−7) = −8
3
The number is −8. Verify: −7 × (−8) = 56 ✓
Exercises
Practice Set
1. Solve: 8x = 56
Divide by 8: x = 56/8 = 7. Verify: 8(7) = 56 ✓
2. Solve: −3x = 18
Divide by −3: x = 18/(−3) = −6. Verify: −3(−6) = 18 ✓
3. Solve: −5x = −45
Divide by −5: x = (−45)/(−5) = 9. Verify: −5(9) = −45 ✓
4. Solve: x / 8 = −4
Multiply by 8: x = −4 × 8 = −32. Verify: −32/8 = −4 ✓
5. Solve: (2/3)x = 14
Multiply by reciprocal 3/2: x = 14 × (3/2) = 42/2 = 21. Verify: (2/3)(21) = 42/3 = 14 ✓
6. Solve: 0.4x = 10
Divide by 0.4: x = 10 / 0.4 = 25. (Or: 0.4 = 2/5, so multiply by 5/2: x = 10 × 5/2 = 25.) Verify: 0.4 × 25 = 10 ✓
7. A car travels a fixed distance. At 60 km/h the journey takes t hours. The distance is 210 km. Find t using the equation 60t = 210.
Divide by 60: t = 210/60 = 3.5. The journey takes 3.5 hours (3 hours 30 minutes). Verify: 60 × 3.5 = 210 ✓
8. Challenge: Solve: (−3/7)x = −9/14
Multiply by reciprocal −7/3: x = (−9/14) × (−7/3) = (9 × 7)/(14 × 3) = 63/42 = 3/2. Verify: (−3/7)(3/2) = −9/14 ✓
Summary
Lesson Checklist
You Can Now
- Solve ax = b by dividing both sides by a (the coefficient of x)
- Solve x/a = b by multiplying both sides by a
- Handle negative coefficients and track the sign of the solution correctly
- Solve equations with fractional coefficients using the reciprocal method
- Solve equations with decimal coefficients
- Verify every solution by substituting back into the original equation
- Translate and solve one-step multiplicative word problems
Stage II Progress — Lessons 16–18
| # | Lesson | Core Skill |
|---|---|---|
| 16 | What is an Equation? | Structure, verification, balance model |
| 17 | One-Step Equations (+ and −) | Subtraction and addition as inverses |
| 18 | One-Step Equations (× and ÷) | Division and multiplication as inverses |
Combine addition/subtraction and multiplication/division into a two-step sequence — and establish the order in which inverse operations must be applied.