Two layers of wrapping. Two inverse operations. One variable freed.
Section 01
Why Two Steps?
Most real equations require more than one inverse operation to isolate the variable. A two-step equation has the general form ax + b = c — the variable has been multiplied by a coefficient and had a constant added or subtracted. Both layers must be removed.
ax + b = c
The critical question is: which layer do you remove first? The answer is always the reverse of BODMAS — you undo operations in the opposite order from how they were applied.
1st
Undo + or − (addition / subtraction)
→
2nd
Undo × or ÷ (mult / division)
Imagine the variable was dressed first in a multiplication (an inner layer), then in an addition (an outer layer). To undress it, you remove the outer layer first — the addition — then the inner layer — the multiplication. Always work from the outside in.
Section 02
The Standard Form: ax + b = c
Worked Example A — Solve 3x + 5 = 17
3x + 5=17
Step 1 — remove the constant (subtract 5)
3x + 5 − 5=17 − 5subtract 5 from both sides
3x=12
Step 2 — remove the coefficient (divide by 3)
3x / 3=12 / 3divide both sides by 3
x=4
Verify
LHS: 3(4) + 5 = 12 + 5 = 17 = RHS ✓
Worked Example B — Solve 4x − 7 = 21
4x − 7=21
Step 1 — add 7 to both sides
4x=28
Step 2 — divide by 4
x=7
Verify
LHS: 4(7) − 7 = 28 − 7 = 21 = RHS ✓
Section 03
Negative Coefficients and Negative Results
Worked Example C — Solve −2x + 9 = 3
−2x + 9=3
Step 1 — subtract 9
−2x=−6
Step 2 — divide by −2
x=3(−6) ÷ (−2) = +3
Verify
LHS: −2(3) + 9 = −6 + 9 = 3 = RHS ✓
Worked Example D — Solve 5 − 3x = 20
5 − 3x=20
Step 1 — subtract 5 from both sides
−3x=15
Step 2 — divide by −3
x=−515 ÷ (−3) = −5
Verify
LHS: 5 − 3(−5) = 5 + 15 = 20 = RHS ✓
Section 04
Two-Step Equations with Fractions and Decimals
Worked Example E — Solve x/4 + 3 = 8
x/4 + 3=8
Step 1 — subtract 3
x/4=5
Step 2 — multiply by 4
x=20
Verify
LHS: 20/4 + 3 = 5 + 3 = 8 = RHS ✓
Worked Example F — Solve 0.5x − 2.4 = 3.6
Step 1 — add 2.4 to both sides
0.5x=6.0
Step 2 — divide by 0.5
x=126 ÷ 0.5 = 12
Verify
LHS: 0.5(12) − 2.4 = 6 − 2.4 = 3.6 = RHS ✓
Section 05
Two-Step Word Problems
Word Problem A — Ayasha thinks of a number, multiplies it by 6, then subtracts 11. The result is 37. Find her number.
1
Let n = the number
2
Equation: 6n − 11 = 37
3
Add 11: 6n = 48
4
Divide by 6: n = 8
5
Ayasha’s number is 8. Verify: 6(8) − 11 = 48 − 11 = 37 ✓
Word Problem B — A plumber charges a R350 call-out fee plus R180 per hour. A customer’s bill was R1 070. How many hours did the plumber work?
1
Let h = hours worked
2
Equation: 180h + 350 = 1070
3
Subtract 350: 180h = 720
4
Divide by 180: h = 4
5
The plumber worked 4 hours. Verify: 180(4) + 350 = 720 + 350 = 1 070 ✓
Section 06
Common Mistakes
Wrong Order — The Classic Error
Dividing before removing the constant is the most frequent two-step mistake:
✗ Wrong — solving 3x + 5 = 17:
Divide by 3 first: x + 5/3 = 17/3 → x = 17/3 − 5/3 = 12/3 = 4
This actually gives the right answer here — but only by accident, and it forces awkward fraction arithmetic. In harder equations it derails entirely. Always remove the constant first.
✓ Correct order: Subtract 5 → divide by 3.
Forgetting to Apply the Operation to Both Sides
✗4x − 7 = 21 → 4x = 21 (forgot to add 7 to the RHS)
7. A school buys identical chairs for each of its 8 classrooms and also one teacher’s desk for R1 200. The total spend is R10 200. Write and solve an equation to find the cost of each chair if every classroom gets 5 chairs.
Total chairs = 8 × 5 = 40. Let c = cost per chair. Equation: 40c + 1200 = 10200. Subtract 1200: 40c = 9000. Divide by 40: c = R225 per chair. Verify: 40(225) + 1200 = 9000 + 1200 = 10 200 ✓
8. Challenge: Solve: (2/3)x − 5/6 = 3/2
Add 5/6 to both sides: (2/3)x = 3/2 + 5/6. LCD = 6: 3/2 = 9/6, so 9/6 + 5/6 = 14/6 = 7/3. Now (2/3)x = 7/3. Multiply by 3/2: x = (7/3) × (3/2) = 21/6 = 7/2. Verify: (2/3)(7/2) − 5/6 = 14/6 − 5/6 = 9/6 = 3/2 ✓
Summary
Lesson Checklist
You Can Now
Identify the two operations applied to the variable in ax + b = c
Apply inverse operations in the correct order: constant first, coefficient second
Solve two-step equations with positive and negative coefficients
Handle equations where the constant is on the left: b − ax = c
Solve two-step equations with fractional and decimal coefficients
Translate and solve two-step word problems
Verify every solution and identify the two most common errors
Up Next → Lesson 20
Multi-Step Equations
Extend the two-step method to equations requiring simplification first — collecting like terms, distributing, and combining across multiple operations before the variable can be isolated.