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Grade 7 / Middle 1 (age 12-13)

Linear Expression Calculation

Linear Expression Calculation

Working with linear expressions is really just tidying up by grouping terms of the same kind. Just as 3 apples plus 2 apples make 5 while apples and oranges cannot be added, 3x and 5x combine but 3x and the constant 2 stay apart. With brackets, the distributive law spreads the multiplier over each term before you add the coefficients of like terms into ax + b. Here you gather square and circle pieces by shape and see from an area picture why distribution works.

Combining Like Terms
2
1
👀 What Are Like Terms?
①Same letter and same degree: 3x and 5x are like terms
②Constants among themselves are like terms: 2 and -7
③3x and 3x² are NOT like terms (different degree)
Distributive Law
3
🔍 How Distribution Works
①k(a + b) = ka + kb
②Multiply each term inside the parentheses by k
③Think of it as a rectangle's area: whole = sum of parts
④Watch the sign carefully when k is negative!
Adding and Subtracting Linear Expressions
Addition of Linear Expressions
(ax + b) + (cx + d) = (a+c)x + (b+d)
Add coefficients of like terms
Subtraction of Linear Expressions
(ax + b) - (cx + d) = (a-c)x + (b-d)
Subtraction is adding the negative
💡 Computation Order
①Remove parentheses (mind signs)
②Group like terms
③Combine coefficients
④Write the final answer as ax + b
Work It Out
Example 1
Simplify 3x + 5 − x + 2.
1
Group like terms.
(3x − x) + (5 + 2)
2
Compute.
= 2x + 7
▸ 2x + 7
Add the coefficients of like terms (same letter).
Example 2
Simplify 2(3x − 1) − (x − 4).
1
Expand the brackets with the distributive law.
6x − 2 − x + 4
2
Collect like terms.
= 5x + 2
▸ 5x + 2
A minus before a bracket flips every sign inside.
Exam Key Points
Core of Linear-Expression Calculation
Combine like terms; use distribution to remove parentheses
Always identify like terms first, then simplify
Grade-7 school exam type
After simplifying 3(2x − 1) + 2(x + 4), what is the sum of the coefficient of x and the constant term?
①11
②12
③13
④14
⑤15
▸ ③ 13
1
Expand with the distributive law.
6x − 3 + 2x + 8
2
It simplifies to 8x + 5, so add coefficient 8 and constant 5.
8x + 5 ⇒ 8 + 5 = 13
🎯 Exam Key Points
①Like terms: same letter AND same degree
②A "-" in front of parentheses flips every sign inside
③Mind sign errors when multiplying by a negative
④Final form: ax + b (descending degree)
⑤0x = 0; if the x coefficient is 0, only the constant remains
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Algebraic Expressions
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Linear Equations
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