Like and unlike terms — simplification

Overview

When we combine algebraic expressions, we can only directly add or subtract terms that have the exact same variable part. This is the foundation of algebraic simplification and appears everywhere from solving equations to calculus.

Figure — Like and unlike terms — simplification

Core Concepts

Why does this work? Derivation from distributive property

Start with the distributive law from arithmetic: $a \cdot c + b \cdot c = (a + b) \cdot c$

Apply to algebra: $3x + 5x = (3 + 5) \cdot x = 8x$

The variable $x$ is the common factor. We're really counting "how many $x$ 's" total.

But with unlike terms: $3x + 5y = 3x + 5y$ No common factor to pull out, so we cannot simplify further.

Worked Examples

Worked example Example 2: Mixed terms

Simplify: $5x^2 + 3x - 2x^2 + 7x - 4$

Solution: Step 1: Group like terms (by variable part)

$x^2$ terms: $5x^2$ and $-2x^2$
$x$ terms: $3x$ and $7x$
Constant terms: $-4$

WHY group? We can only combine terms with identical variable parts. Grouping makes it visible.

Step 2: Combine each group

$x^2$ terms: $5x^2 - 2x^2 = (5-2)x^2 = 3x^2$
$x$ terms: $3x + 7x = (3+7)x = 10x$
Constants: $-4$ (nothing to combine with)

WHY keep separate? $x^2$ , $x$ , and constants are all unlike to each other.

Step 3: Write in standard form (descending powers) $= 3x^2 + 10x - 4$

Answer: $3x^2 + 10x - 4$

Common Mistakes

Active Recall Practice

Recall Feynman Test: Explain to a 12-year-old

Imagine you're collecting fruits. You have 3 apples and 5 aples — you can combine them to get 8 apples because they're the same type of fruit.

But if you have 3 aples and 5 oranges, you can't say you have "8 apple-oranges" or something weird. You just have 3 apples and 5 oranges separately.

In algebra, $x$ and $y$ are like different fruits. $3x$ and $5x$ are the same type (both are " $x$ -fruits"), so we can combine them: $3x + 5x = 8x$ . But $3x$ and $5y$ are different types, so they stay separate: $3x + 5y$ cannot be simplified.

The rule: Only combine terms that have exactly the same letter-part!

Connections

2.1.01-Variables-constants-and-coefficients — You need to identify coefficients and variables before combining
2.1.03-Addition-and-subtraction-of-algebraic-expressions — Uses like term combining as the core skill
2.2.01-Distributive-property — The theoretical foundation for why like terms combine
3.1.01-Solving-linear-equations-one-variable — Every equation solution requires simplifying like terms
4.1.01-Polynomials-introduction — Polynomial simplification is just like terms on steroids

Flashcards

#flashcards/maths

What are like terms? :: Terms with identical variable parts (same letters raised to same powers), differing only in coefficients

Can you add

5x^2

and

3x^3

? Why or why not?

No, they are unlike terms because the powers of

x

are different (

x^2

x^3

)

Simplify:

7a + 3b - 2a + 5b

5a + 8b

(combine like terms:

7a - 2a = 5a

and

3b + 5b = 8b

)

What stays the same when combining like terms?

The variable part stays unchanged; only coefficients are added/subtracted

Why can't we simplify

4x + 3y

Because

x

and

y

are different variables, making them unlike terms with no common factor to combine

Simplify:

5xy + 2x + 3xy - x

8xy + x

(like terms:

5xy + 3xy = 8xy

and

2x - x = x

)

What is the coefficient when combining

-3x^2 + 7x^2

4

(the combined term is

4x^2

because

-3 + 7 = 4

)

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Like aur Unlike Terms ka Concept:

Jab bhi hum algebra mein expressions simplify karte hain, sabse pehle yeh dekhna padta hai ki kaunse terms "like" hain aur kaunse "unlike". Iska matlab simple hai —agar do terms mein variable part bilkul same hai (same letter, same power), toh woh like terms hain aur hum unhe directly add ya subtract karakte hain. For example, 3x aur 5x dono mein sirf "x" hai, toh yeh like terms hain: 3x + 5x = 8x. Lekin agar 3x aur 5y hain, toh yeh unlike hain kyunki ek mein x hai aur dusre mein y — different variables matlab different "types" ki chezein hain, jaise aples aur oranges ko alag count karte hain.

Simplification ka process:

Jab koi bada expression ho jaise 5x² + 3x - 2x² + 7x - 4, toh sabse pehle hum grouping karte hain — matlab same variable part wale terms ko ek sath rakhte hain. Phir har group ko separately combine karte hain: x² wale terms (5x² - 2x²) = 3x², phir x wale terms (3x + 7x) = 10x, aur constants separately (-4). Final answer: 3x² + 10x - 4. Yeh skill bahut zaroori hai kyunki age jake equations solve karne mein, polynomials mein, calculus mein — har jagah like terms combine karne padenge. Galti yeh hoti hai ki students kabhi coefficients add karke variable bhi square kar dete hain (4x + 3x = 7x² — WRONG!), ya phir unlike terms ko combine kar dete hain. Yad rakho: variable part same ho

Like and unlike terms — simplification

Overview

Core Concepts

Why does this work? Derivation from distributive property

Worked Examples

Common Mistakes

Active Recall Practice

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Algebra — Introduction & Intermediate

Connections