LESSON 20 / 100 | 20% COMPLETE | STAGE II — EQUATIONS
Stage II — Equations
Multi-Step Equations
Simplify first, then solve — clearing the path before walking it.
Section 01
When Two Steps Aren’t Enough
Many equations don’t arrive in the clean ax + b = c form of Lesson 19. They contain like terms that need collecting, brackets that need expanding, or constants scattered on both sides. Before you can apply inverse operations, you must simplify.
Master Strategy — Multi-Step Equations
1
Expand any brackets (distributive property)
2
Collect like terms on each side separately
3
Move variable terms to one side, constants to the other
4
Isolate the variable — divide or multiply as the final step
5
Verify by substituting back into the original equation
A multi-step equation is a room that needs tidying before you can find what you’re looking for. Expand the brackets (open the cupboards), collect like terms (group similar items), move constants (clear the floor), then isolate x (find the thing you need).
Section 02
Collect Like Terms First
Worked Example A — Solve 5x + 3 + 2x − 1 = 30
5x + 3 + 2x − 1
=
30
Step 1 — collect like terms on the LHS
7x + 2
=
30
5x + 2x = 7x; 3 − 1 = 2
Step 2 — subtract 2
7x
=
28
Step 3 — divide by 7
x
=
4
Verify
LHS: 5(4) + 3 + 2(4) − 1 = 20 + 3 + 8 − 1 = 30 = RHS ✓
Section 03
Variables on Both Sides
When variable terms appear on both sides, move them to one side by adding or subtracting the variable term. Convention is to gather variables on the left and constants on the right — though either side works.
Worked Example B — Solve 7x − 4 = 3x + 12
7x − 4
=
3x + 12
Step 1 — subtract 3x from both sides
4x − 4
=
12
7x − 3x = 4x
Step 2 — add 4 to both sides
4x
=
16
Step 3 — divide by 4
x
=
4
Verify
LHS: 7(4) − 4 = 24. RHS: 3(4) + 12 = 24. LHS = RHS ✓
Worked Example C — Solve 9 − 2x = 5x − 12
9 − 2x
=
5x − 12
Step 1 — add 2x to both sides
9
=
7x − 12
Step 2 — add 12 to both sides
21
=
7x
Step 3 — divide by 7
3
=
x → x = 3
Verify
LHS: 9 − 2(3) = 3. RHS: 5(3) − 12 = 3. LHS = RHS ✓
Section 04
Combining All Steps — Full Multi-Step
Worked Example D — Solve 4x + 3 − x = 2x − 9 + 3x
4x + 3 − x
=
2x − 9 + 3x
Step 1 — collect like terms on each side
3x + 3
=
5x − 9
LHS: 4x−x=3x. RHS: 2x+3x=5x
Step 2 — subtract 3x from both sides
3
=
2x − 9
Step 3 — add 9 to both sides
12
=
2x
Step 4 — divide by 2
x
=
6
Verify
LHS: 4(6) + 3 − 6 = 21. RHS: 2(6) − 9 + 3(6) = 12 − 9 + 18 = 21. LHS = RHS ✓
Section 05
Multi-Step Word Problems
Word Problem — Three consecutive even integers sum to 78. Find them.
1
Let first even integer = n. Then the next two are n + 2 and n + 4.
2
Equation: n + (n + 2) + (n + 4) = 78
3
Collect: 3n + 6 = 78
4
Subtract 6: 3n = 72
5
Divide by 3: n = 24
6
The integers are 24, 26, and 28. Verify: 24 + 26 + 28 = 78 ✓
Section 06
When Solving Reveals No Solution or All Solutions
Multi-step equations sometimes simplify to a statement with no variable left. The result tells you what kind of equation you have.
Identity — Solve 3(x + 2) = 3x + 6
1
Expand: 3x + 6 = 3x + 6
2
Subtract 3x: 6 = 6
3
Always true — all real numbers are solutions.
Contradiction — Solve 2x + 5 = 2x + 9
1
Subtract 2x: 5 = 9
2
Never true — no solution exists.
Exercises
Practice Set
1. Solve: 3x + 5 + 2x − 3 = 27
Collect: 5x + 2 = 27. Subtract 2: 5x = 25. Divide by 5: x = 5. Verify: 3(5)+5+2(5)−3 = 27 ✓
2. Solve: 8x − 3 = 5x + 9
Subtract 5x: 3x − 3 = 9. Add 3: 3x = 12. Divide by 3: x = 4. Verify: 8(4)−3 = 29; 5(4)+9 = 29 ✓
3. Solve: 12 − 3x = 4x − 9
Add 3x: 12 = 7x − 9. Add 9: 21 = 7x. Divide: x = 3. Verify: 12−9=3; 4(3)−9=3 ✓
4. Solve: 6x + 4 − 2x = 3x − 1 + x
Collect each side: 4x + 4 = 4x − 1. Subtract 4x: 4 = −1. Contradiction — no solution.
5. Solve: 5(x − 1) + 2x = 3(x + 5) − 2 (expand first)
Expand: 5x − 5 + 2x = 3x + 15 − 2. Collect: 7x − 5 = 3x + 13. Subtract 3x: 4x − 5 = 13. Add 5: 4x = 18. Divide: x = 4.5. Verify: LHS = 7(4.5)−5 = 26.5; RHS = 3(4.5)+13 = 26.5 ✓
6. Three consecutive odd integers sum to 51. Find them.
Let n = first odd integer. Equation: n + (n+2) + (n+4) = 51. 3n + 6 = 51. 3n = 45. n = 15. Integers: 15, 17, 19. Verify: 15+17+19 = 51 ✓
7. Solve: 2(3x − 4) = 4(x + 1) − 2
Expand: 6x − 8 = 4x + 4 − 2. Collect RHS: 6x − 8 = 4x + 2. Subtract 4x: 2x − 8 = 2. Add 8: 2x = 10. x = 5. Verify: LHS = 2(11)=22; RHS = 4(6)−2=22 ✓
8. Challenge: Solve: 3(2x + 1) − 2(x − 4) = 4(x + 3) + 1
Expand: 6x + 3 − 2x + 8 = 4x + 12 + 1. Collect: 4x + 11 = 4x + 13. Subtract 4x: 11 = 13. Contradiction — no solution. (The equation is inconsistent regardless of x.)
Summary
Lesson Checklist
You Can Now
- Apply the 5-step master strategy: expand → collect → move variables → move constants → isolate
- Simplify both sides by collecting like terms before solving
- Move variable terms across the equals sign by adding or subtracting
- Solve equations requiring three or more steps
- Recognise identities (always true) and contradictions (never true) from multi-step solving
- Translate and solve multi-step word problems including consecutive integer problems
Focus entirely on the expansion step — mastering equations where brackets are the primary obstacle and distributing correctly is the key that unlocks the solution.