Lesson 5 Unit 2 · Patterns & Algebra
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.
Find out
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Simplify $(5x + 3) + (2x - 7)$. Show every step and explain what happens to each type of term.
1
The Big Idea
Adding and subtracting algebraic expressions is just collecting like terms with extra steps. When you see brackets, remove them first , then combine what matches.
$(3x + 5) + (2x - 3)$ means adding all terms: $3x + 5 + 2x - 3$ . $x$-terms: $3x + 2x = 5x$. Constants: $5 - 3 = 2$. Answer: $5x + 2$.
$(3x+5)$ Expression 1
+
$(2x-3)$ Expression 2
$3x + 5 + 2x - 3$
$5x + 2$
SIMPLIFIED
$(3x+5) + (2x-3) = 5x + 2$
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.
2
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
3
Words You Need
vocabulary
Add Expressions Combine two or more expressions by removing brackets and collecting like terms.
Subtract Expressions Remove brackets, flip signs of terms being subtracted, then collect like terms.
Expand Brackets Remove brackets by distributing or by dropping when preceded by + or -.
Distribute Multiply a term outside brackets by every term inside. e.g. $2(x+3) = 2x+6$.
Column Method Writing like terms in columns to add/subtract them neatly, like column arithmetic.
Simplified An expression with no like terms left to combine. e.g. $5x + 2$ is fully simplified.
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.
5
The Subtraction Rule
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: $3x + 7$ .
$(5x+3)$ keeps signs
-
$(2x-4)$ SIGNS FLIP
$2x$ → $-2x$
$-4$ → $+4$
All terms flip sign
$= 5x + 3 - 2x + 4$
$= 3x + 7$
$-(2x - 4) = -2x + 4$
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.
Write like terms in columns and add them vertically. It's just like primary school column addition, but with letters attached.
ADD
$3x + 2y + 5$
$5x - 3y + 2$
$8x - y + 7$
$x$: $3+5=8$
$y$: $2+(-3)=-1$
const: $5+2=7$
SUBTRACT
$3x + 2y + 5$
$5x - 3y + 2$
$-2x + 5y + 3$
$x$: $3-5=-2$
$y$: $2-(-3)=5$
const: $5-2=3$
Signs FLIP for subtraction!
7
Distributing a Multiplier
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 .
$3$
$(2x+4)$
Multiply 3 by each term inside
$3(2x)$ $= 6x$
+
$3(4)$ $= 12$
$6x + 12$
$3(2x+4) = 3(2x) + 3(4) = 6x + 12$
Expression Distribution Result
$2(x+3)$ $2x + 6$ $2x + 6$ $4(3a-2)$ $12a - 8$ $12a - 8$ $-(2x+5)$ $-2x - 5$ $-2x - 5$ $-(x-3)$ $-x + 3$ $-x + 3$
8
Multi-Step Simplification
Some problems have both addition and distribution . Remove brackets first, then collect like terms. Work systematically.
$(3x + 2) + 2(x - 1)$ Original expression
$= 3x + 2 + 2x - 2$ Drop first bracket + distribute the 2
$= 5x$ Collect like terms: $3x+2x=5x$ and $+2-2=0$
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
Q
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: $5(1) - 3 = 2$ ✓
Show me the next step
Answer $5x - 3$
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.
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Brain Trainer · 4 problems
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Brain Trainer · Mixed
4 problems
1 Simplify $(4x + 3) + (2x - 5)$.
Show answer $4x + 3 + 2x - 5 = 6x - 2$$6x - 2$
2 Simplify $(3a + 7) - (a - 2)$.
Show answer $3a + 7 - a + 2 = 2a + 9$. Watch the $-(-2)$ become $+2$!$2a + 9$
3 Simplify $3(2x + 1) - 4$.
Show answer $6x + 3 - 4 = 6x - 1$$6x - 1$
4 Simplify $2(x + 3) + (4x - 1)$.
Show answer $2x + 6 + 4x - 1 = 6x + 5$$6x + 5$
$(5x + 3) + (2x - 1) = $
A OPTION A $7x + 4$ B OPTION B $7x + 2$ C OPTION C $7x + 2$ D OPTION D $3x + 2$
$(4x + 5) - (2x - 3) = $
A OPTION A $2x + 2$ B OPTION B $2x + 8$ C OPTION C $6x + 8$ D OPTION D $2x - 8$
$2(3x + 4) = $
A OPTION A $6x + 8$ B OPTION B $6x + 4$ C OPTION C $5x + 8$ D OPTION D $3x + 8$
Simplify $2(x + 3) - (x - 2)$.
A OPTION A $x + 5$ B OPTION B $3x + 1$ C OPTION C $x + 8$ D OPTION D $x + 8$
A student writes $(3x+4) - (2x-1) = 3x+4-2x-1 = x+3$. What is the mistake?
A OPTION A They forgot to subtract $2x$ B OPTION B They flipped the sign of $2x$ but not $-1$ C OPTION C They didn't remove the brackets D OPTION D The answer is actually correct
Show Your Working · 3 questions
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9 marks total
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Medium
2 MARKS
Q6. Simplify $(6a + 2) + (3a - 5)$.
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Medium
3 MARKS
Q7. Simplify $3(2x + 1) - 2(x - 4)$.
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Medium
4 MARKS
Q8. The perimeter of a shape is $(3x+2)+(5x-1)+(2x+4)$. Simplify and find the perimeter when $x=3$.
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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.
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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.
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