Linear Equations · Grade 7
How do you collect like terms and expand brackets?
Like terms carry the same letter to the same power, and only those can be combined: three x plus two x is five x, but three x plus two stays as it is. The distributive law says to multiply whatever stands before the brackets by every term inside them, skipping none.
Learning objectives
- Identify like terms inside an expression and collect them
- Open brackets with the distributive law, including when a minus sign stands in front
- Simplify an expression that contains both brackets and like terms
What may be combined
Three x plus two x is five x, exactly as three apples plus two apples is five apples. The x is the name of the unit, and counting does not change it.
Three x plus two is not five of anything. Those are two different units, and the expression simply stays as it is — that is a complete answer, not a partial one.
The coefficient is the number in front of the letter, and the sign belongs to it. In minus four x the coefficient is minus four, and collecting must carry that sign along.
The distributive law
Whatever stands before the brackets multiplies every term inside them, without exception. Three times the bracket x plus two gives three x plus six.
A quick count prevents the usual error: the number of terms in the result equals the number inside the brackets. If two went in and one came out, a term was dropped.
A minus before the brackets
A minus before brackets is multiplication by minus one, so it flips the sign of every term inside. Minus the bracket x minus three gives minus x plus three.
The second term is the one that gets missed. The first sign almost always flips and the second is forgotten, so it is worth writing the expansion step out rather than doing it in your head.
Worked examples
Collect like terms: 5x plus 3 minus 2x plus 4
- Mark the x terms: 5x and minus 2x
- Collect them: 5 minus 2 gives 3x
- Collect the numbers: 3 plus 4 gives 7
Answer: 3x plus 7
Expand: 4(x − 3)
- Multiply 4 by the first term: 4x
- Multiply 4 by the second, with its sign: minus 12
- Two terms went in, two came out
Answer: 4x minus 12
Simplify: 2(x + 5) minus (x − 1)
- Expand the first: 2x plus 10
- The minus flips both signs: minus x plus 1
- Collect the x terms: 2x minus x gives x
- Collect the numbers: 10 plus 1
Answer: x plus 11
Common mistakes
- Combining an x term with a plain number
- Those are two different units. Three x plus two is neither five x nor five — the expression stays as it is, and that is the answer.
- Multiplying only the first term inside the brackets
- The distributive law reaches every term. Counting the terms before and after the expansion catches the missing one in a second.
- Flipping only one sign after a minus
- A minus before brackets is multiplication by minus one, and it applies to everything inside. The second term is the one that is nearly always forgotten.
What to remember
- Only terms of the same kind combine.
- The sign is part of the coefficient.
- Distributing reaches every term in the brackets.
- A minus before brackets flips every sign.