Calculating with linear expressions is a matter of gathering like with like.
Just as 3 apples and 2 apples make 5 apples but apples and oranges cannot be counted together, 3x and 2x combine to 5x while 3x and 2 stay separate.
If there are brackets, use the distributive law to multiply each term inside by the number in front, then combine the coefficients of like terms to reach the form ax + b.
On this page you gather tiles of the same shape, and see how an area picture splits depending on the multiplier.
Collecting like terms
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3x + 2 with more x terms and constants added, then gathered by shape
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👀 Like terms
①Terms with the same letters to the same powers are like terms: 3x and 5x are like terms
②Constant terms are like terms too, such as 2 and −7
③3x and 3x² have the same letter but different powers, so they are not like terms; nor are 3x and 3y
④In the picture, each □ stands for one x and each ○ for one 1. You can only count same shapes together, and a red −x meeting a gold x makes 0 and disappears
The distributive law
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The distributive law as area, k × (2x + 3): the 2x part and the 3 part are each multiplied by k
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🔍 The distributive law
①k × (a + b) = k × a + k × b: multiply every term inside the brackets by k
②As area, it is a rectangle made of k strips, each 2x + 3 long. The whole area equals the area of the 2x part plus the area of the 3 part
③If k is negative, every term in the brackets changes sign: −2 × (2x + 3) = −4x − 6
④A minus sign alone in front of brackets means multiplying by −1: −(x − 4) = −x + 4
Adding and subtracting linear expressions
Adding linear expressions
(ax + b) + (cx + d) = (a + c)x + (b + d)
Remove the brackets and add the coefficients of like terms
Subtracting linear expressions
(ax + b) − (cx + d) = (a − c)x + (b − d)
Change the sign of every term being subtracted, then add
💡 Order of work
①Expand brackets with the distributive law; a negative in front changes every sign inside
②Gather like terms
③Combine the coefficients of like terms
④Write the x term first and the constant last, in the form ax + b
Check with examples
Example 1
Simplify 3x + 5 − x + 2.
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Gather like terms.
(3x − x) + (5 + 2)
2
Combine the coefficients.
= 2x + 7
▸ 2x + 7
With tiles, three x and one −x leave two x, and five 1s and two 1s make seven.
Example 2
Simplify 2(3x − 1) − (x − 4).
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Expand the brackets. The second bracket is multiplied by −1.
6x − 2 − x + 4
2
Gather like terms.
= 5x + 2
▸ 5x + 2
−(x − 4) becomes −x + 4. Leaving the −4 as −4 is the most common slip.
Exam checklist
Linear expressions in short
expand brackets → gather like terms → combine coefficients
First pick out which terms are alike, then tidy up
Grade 7 school test style
When 3(2x − 1) + 2(x + 4) is simplified, what is the sum of the x coefficient and the constant term?
①11
②12
③13
④14
⑤15
▸ ③ 13
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Expand the brackets.
6x − 3 + 2x + 8
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It simplifies to 8x + 5, so add the coefficient 8 and the constant 5.
8x + 5 ⇒ 8 + 5 = 13
🎯 Exam checklist
①Like terms must have the same letters to the same powers
②A minus sign in front of brackets changes the sign of every term inside
③When distributing a negative, it is easy to forget the sign of the last term
④Write the answer as ax + b, with the x term first
⑤If the x coefficient becomes 0, only the constant is left