Simplifying · Three Variables · D5

Three-variable mastery.

Bring it all together across three variables: distribute, collect each letter, keep signs.

01 / THE IDEA

See it simplify.

Watch it simplify

Three-variable mastery

Distribute, then collect x, y, z with their signs.

1 / 3
3(x+y)+2(y+z)+x3(x + y) + 2(y + z) + x
1

Distribute

Multiply out the brackets — every term inside by the factor in front.

3x+3y+2y+2z+x3x + 3y + 2y + 2z + x
2

Group like terms

Gather the terms that share the same variable part.

(3x+x)+(3y+2y)+2z(3x + x) + (3y + 2y) + 2z
3

Combine

Add the coefficients within each group.

4x+5y+2z4x + 5y + 2z
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Simplify it.

Type the simplest form (e.g. 6x + 5). Use ^ for powers.

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— / WHY IT WORKS

The idea underneath.

Three-variable method

Expand → collect xs → ys → zs → order x, y, z.

Why it works

Three colour piles, sorted clean.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

4x + 4y + 3z + 2x

1

Group like terms

Gather the terms that share the same variable part. → (4x + 2x) + 4y + 3z

2

Combine

Add the coefficients within each group. → 6x + 4y + 3z

EX 2medium

12x + 4y - 7x + 4z + 7y

1

Group like terms

Gather the terms that share the same variable part. → (12x - 7x) + (4y + 7y) + 4z

2

Combine

Add the coefficients within each group. → 5x + 11y + 4z

EX 3hard

6x + 11y + 4z + 5x

1

Group like terms

Gather the terms that share the same variable part. → (6x + 5x) + 11y + 4z

2

Combine

Add the coefficients within each group. → 11x + 11y + 4z

04 / COMMON TRAPS

Where students slip.

Trap

Sign or bracket slips.

Fix

Go slowly: expand, then collect.

Trap

Half-collected answers.

Fix

Combine fully.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Simplify 4x + 4y + 3z + 2x

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
C2

Simplify 6x + 4y - 2x + 5z + 5y

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
C3

Simplify 8x + 6y - 4x + 4z + 6y

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
C4

Simplify 5(x + y) + 5(y + z) + 3x

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
Apply

Apply (4)

A1

Simplify 12x + 4y - 7x + 4z + 7y

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
A2

Simplify 5x + 7y - 4x + 6z + 4y

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
A3

Simplify 7x + 5y + 7z + 4x

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
A4

Simplify 6x + 4y - 3x + 3z + 9y

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
Extension

Extension (4)

E1

Simplify 6x + 11y + 4z + 5x

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
E2

Simplify 6(x + y) + 9(y + z) + 8x

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
E3

Simplify 4x + 4y + 6z + 10x

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.
E4

Simplify 13x + 9y - 9x + 9z + 7y

Need a hint?
  • Distribute, then collect each letter.
  • Order x, y, z.