Lesson 5~30 minUnit 2 · Patterns & Algebra+90 XP

Adding and Subtracting Algebraic Terms

Now that you know what like terms are, master the arithmetic of combining them. Column addition, subtraction with negatives, and multi-step simplification.

Today's hook: If you owe 3 dollars and earn 8 dollars, what's your balance? That's $-3 + 8 = 5$, and it's the same for algebraic terms.

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Think First

warm-up

Simplify $(5x + 3) + (2x - 7)$. Show every step and explain what happens to each type of term.

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The Big Idea

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Adding and subtracting algebraic expressions is just collecting like terms with extra steps. When you see brackets, remove them first, then combine what matches.

Brackets and other grouping symbols mark the terms that must be treated as one group. A plus sign before a group preserves every sign, while a minus sign means subtract the whole group, so $(3x+2)-(x-4)=3x+2-x+4$.

$(3x + 5) + (2x - 3)$ means adding all terms: $3x + 5 + 2x - 3$. $x$-terms: $3x + 2x = 5x$. Constants: $5 - 3 = 2$. Answer: $5x + 2$.

$(3x+5) + (2x-3) = 5x + 2$

Remove brackets first

$(a+b) + (c+d)$ becomes $a+b+c+d$. Drop the brackets and combine.

Signs travel with terms

In $(2x-3)$, the $-3$ keeps its minus sign when you remove brackets.

Then collect like terms

Group $x$ terms and constants separately, just like in the last lesson.

What You'll Master

objectives

Know

How to add algebraic expressions
How to subtract algebraic expressions
That brackets must be removed first

Understand

Why subtraction flips all signs in the bracket
That adding brackets preserves signs
How to check answers by substitution

Can Do

Add and subtract expressions with brackets
Handle multiple variables
Simplify complex multi-step expressions

Words You Need

vocabulary

Add ExpressionsCombine two or more expressions by removing brackets and collecting like terms.

Subtract ExpressionsRemove brackets, flip signs of terms being subtracted, then collect like terms.

Expand BracketsRemove brackets by distributing or by dropping when preceded by + or -.

DistributeMultiply a term outside brackets by every term inside. e.g. $2(x+3) = 2x+6$.

Column MethodWriting like terms in columns to add/subtract them neatly, like column arithmetic.

SimplifiedAn expression with no like terms left to combine. e.g. $5x + 2$ is fully simplified.

Spot the Trap

heads-up

Wrong: $(4x + 3) - (2x - 1) = 4x + 3 - 2x - 1$

Right: $= 4x + 3 - 2x + 1$. The $-1$ becomes $+1$ when subtracting! Minus the bracket flips ALL signs.

Wrong: $3(x + 2) = 3x + 2$

Right: $3(x+2) = 3x + 6$. The 3 multiplies both terms inside.

The Subtraction Rule

+5 XP

Subtracting an expression means flipping every sign inside it. Think of it as multiplying everything inside by $-1$.

$(5x + 3) - (2x - 4)$: remove brackets, $5x + 3 - 2x + 4$. The $-4$ becomes $+4$! Then $x$-terms: $5x - 2x = 3x$. Constants: $3 + 4 = 7$. Answer: $3x + 7$.

$-(2x - 4) = -2x + 4$

Minus = flip all

$-(a + b) = -a - b$. Every term changes sign. No exceptions!

Check: put a number in

Let $x=0$: $(5(0)+3) - (2(0)-4) = 3 - (-4) = 7$. Our answer $3(0)+7=7$ ✓.

Double negative = positive

$5 - (-4) = 5 + 4$. The minus flips the negative to positive.

The Column Method

+5 XP

Write like terms in columns and add them vertically. It's just like primary school column addition, but with letters attached.

Distributing a Multiplier

+5 XP

When a number sits outside brackets, multiply it by every term inside. $2(x + 3)$ means two lots of $x$ plus two lots of $3$.

$3(2x + 4)$ = $3 \times 2x + 3 \times 4$ = $6x + 12$. The 3 multiplies both terms. This is called the distributive law.

$3(2x+4) = 3(2x) + 3(4) = 6x + 12$

Expression	Distribution	Result
$2(x+3)$	$2x + 6$	$2x + 6$
$4(3a-2)$	$12a - 8$	$12a - 8$
$-(2x+5)$	$-2x - 5$	$-2x - 5$
$-(x-3)$	$-x + 3$	$-x + 3$

Multi-Step Simplification

+5 XP

Some problems have both addition and distribution. Remove brackets first, then collect like terms. Work systematically.

Try It Now: Simplify $4(2x+1) - (3x-2)$.Answer: $8x+4-3x+2 = 5x+6$

Watch Me Solve It · Worked example

Watch Me Solve It · Full simplification

+15 XP per step

PROBLEM

Simplify $2(3x - 1) - (x + 4) + 3$.

1
Distribute the 2
$2(3x) + 2(-1) = 6x - 2$
The 2 multiplies both $3x$ and $-1$. Every term inside gets multiplied.
2
Remove the subtracted bracket (flip signs)
$-(x+4) = -x - 4$
Minused bracket: $x$ becomes $-x$, $+4$ becomes $-4$. Both signs flip!
3
Rewrite the full expression
$6x - 2 - x - 4 + 3$
Combine all terms. The $+3$ at the end has no bracket, so it stays $+3$.
4
Collect like terms
$x$-terms: $6x - x = 5x$. Constants: $-2 - 4 + 3 = -3$
$6 - 1 = 5$ for the $x$ terms. $-2 - 4 = -6$, then $-6 + 3 = -3$ for constants.
5
Write the final answer
$5x - 3$
Check: let $x=1$. Original: $2(3-1) - (1+4) + 3 = 4 - 5 + 3 = 2$. Answer: $5(1) - 3 = 2$ ✓

Answer$5x - 3$

Common Pitfalls

heads-up

Forgetting to distribute to all terms

$2(x+3) = 2x + 3$ is wrong. The 2 must multiply both $x$ AND $3$.

Fix: count the terms inside the bracket. Multiply the outside number by EACH one.

Not flipping signs for subtraction

$(5x+2) - (3x-1) = 5x+2-3x-1$ is wrong. The $-1$ should become $+1$.

Fix: when subtracting a bracket, flip EVERY sign inside. $-(3x-1) = -3x+1$.

Dropping the variable after combining

$5x - 5x = 5$ is wrong. $5x - 5x = 0$. The terms cancel completely!

Fix: when coefficients cancel (e.g. $5-5$), the answer is $0$, not the coefficient.

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

1 Simplify $(4x + 3) + (2x - 5)$.
$4x + 3 + 2x - 5 = 6x - 2$$6x - 2$
2 Simplify $(3a + 7) - (a - 2)$.
$3a + 7 - a + 2 = 2a + 9$. Watch the $-(-2)$ become $+2$!$2a + 9$
3 Simplify $3(2x + 1) - 4$.
$6x + 3 - 4 = 6x - 1$$6x - 1$
4 Simplify $2(x + 3) + (4x - 1)$.
$2x + 6 + 4x - 1 = 6x + 5$$6x + 5$

Complete in workbook.

Multiple Choice

+10 XP

$(5x + 3) + (2x - 1) = $

Multiple Choice

+10 XP

$(4x + 5) - (2x - 3) = $

Multiple Choice

+10 XP

$2(3x + 4) = $

Multiple Choice

+10 XP

Simplify $2(x + 3) - (x - 2)$.

Multiple Choice

+10 XP

A student writes $(3x+4) - (2x-1) = 3x+4-2x-1 = x+3$. What is the mistake?

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 2 MARKS

Q6. Simplify $(6a + 2) + (3a - 5)$.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. Simplify $3(2x + 1) - 2(x - 4)$.

Answer in your workbook.

Apply Medium 4 MARKS

Q8. The perimeter of a shape is $(3x+2)+(5x-1)+(2x+4)$. Simplify and find the perimeter when $x=3$.

Answer in your workbook.

Apply Hard 3 MARKS

Stretch. Three consecutive even numbers can be written as $x$, $x+2$, $x+4$. Write and simplify an expression for their sum.

Answer in your workbook.

Stretch Challenge · +25 XP, +10 coins

Extension Problems

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Quick Review

Key Concept

Review the main ideas from this lesson.

Formulas

Key formulas and rules.

Watch Out

Common mistakes to avoid.

Check

Always verify your answers.

Practice

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Interactive: Algebra Machine

Substitute numbers into algebraic expressions and see them evaluate step by step.

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Unit overview · Maths subject page · Unit quiz

Adding and Subtracting Algebraic Terms

Printable Worksheets

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Can Do

Extension Problems

Key Concept

Formulas

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Interactive: Algebra Machine

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