Adding and Subtracting Terms
Like terms have exactly the same pronumeral part, such as \(5xy\) and \(-2yx\), and only like terms can be added or subtracted. In NSW Year 8 maths you simplify by combining the coefficients and keeping the pronumeral part as it is (NESA outcome MA4‑ALG‑C‑01).
You will collect like terms in expressions with negatives, squared terms and several pronumerals, such as \(7x^2 + 5x - 6x^2 - 7x\), and leave unlike terms alone. You will use the commutative and associative properties to move each term with its sign, and check answers by substituting a friendly number. Adding two clothing orders for a netball club works in exactly the same way. It is part of Year 8 Algebra.
Theory
Only like terms can be added or subtracted. Like terms have exactly the same pronumeral part: \(3x\) and \(2x\), \(4y^2\) and \(y^2\), \(2xy\) and \(4yx\). Add or subtract the coefficients and keep the pronumeral part. This Year 8 maths page shows how to collect like terms.
Every term has a coefficient (the number in front) and a pronumeral part (the pronumerals with their indices). In \(-7gh\) the coefficient is \(-7\) and the pronumeral part is \(gh\). A term with no number in front has coefficient \(1\): \(v = 1v\) and \(-v = -1v\).
Like terms have the same pronumeral part, pronumeral for pronumeral and index for index. \(3x\) and \(2x\), \(4y^2\) and \(y^2\), and \(2xy\) and \(4yx\) are like terms. \(2x\) and \(3y\), and \(2xy^2\) and \(x^2y\), are not.
The order of multiplying does not matter (the commutative property), so \(yx\) is the same as \(xy\). Terms can also be swapped and regrouped when adding (the commutative and associative properties), which is why like terms can be moved next to each other.
To simplify: group the like terms, moving each term with the sign in front of it; add or subtract the coefficients and keep the pronumeral part, \(9t - 4t = 5t\); leave any term with no like term as it is. Answers are usually written with the highest index first.
Check by substituting a friendly number for each pronumeral: the expression and the answer must give the same value.
Coefficient and pronumeral part:
| Term | Coefficient | Pronumeral part |
|---|---|---|
| \(6k\) | \(6\) | \(k\) |
| \(-k\) | \(-1\) | \(k\) |
| \(6k^2\) | \(6\) | \(k^2\) |
| \(tk\) | \(1\) | \(kt\) |
| \(6\) | \(6\) | none (a constant) |
How to collect like terms
- Find the like terms (same pronumeral part; \(xy = yx\)).
- Group them, each with the sign in front of it.
- Add or subtract the coefficients; keep the pronumeral part.
- Check by substituting numbers.
Group the like terms: \(2x^2 + x^2 + 3x + 4x\).
| \(2x^2 + x^2\) | \(=\) | \(3x^2\) |
| \(3x + 4x\) | \(=\) | \(7x\) |
\(3x^2 + 7x\)
Write \(4yx\) as \(4xy\): \(2xy + 4xy + x + y\).
| \(2xy + 4xy\) | \(=\) | \(6xy\) |
\(6xy + x + y\) (\(yx\) is the same as \(xy\))
(i) \(7t + t - 3t\)
(ii) \(-5w - 4w\)
| \(7t + t - 3t\) | \(=\) | \((7 + 1 - 3)t = 5t\) |
| \(-5w - 4w\) | \(=\) | \((-5 - 4)w = -9w\) |
(i) \(5t\). Check with \(t = 4\): \(28 + 4 - 12 = 20\) and \(5 \times 4 = 20\).
(ii) \(-9w\)
| \(\) | \(=\) | \(9g - 4g + 2h - 7 + 5\) |
| \(\) | \(=\) | \(5g + 2h - 2\) |
\(5g + 2h - 2\). Check with \(g = 3\), \(h = 4\): \(27 - 7 - 12 + 8 + 5 = 21\) and \(15 + 8 - 2 = 21\).
| \(4k^2 - k^2\) | \(=\) | \(3k^2\) |
| \(3kv - 5kv\) | \(=\) | \(-2kv\) |
\(3k^2 - 2kv\). Check with \(k = 2\), \(v = 5\): \(16 + 30 - 4 - 50 = -8\) and \(12 - 20 = -8\).
| \(6n^2 - n^2\) | \(=\) | \(5n^2\) |
| \(-5n + 8n\) | \(=\) | \(3n\) |
| \(2 - 9\) | \(=\) | \(-7\) |
\(5n^2 + 3n - 7\). Check with \(n = 2\): \(24 - 10 + 2 - 4 + 16 - 9 = 19\) and \(20 + 6 - 7 = 19\).
Common pitfalls
Frequently asked questions
What are like terms?
Terms with exactly the same pronumeral part, such as \(3ab\) and \(-5ba\).
How do you add like terms?
Add the coefficients and keep the pronumeral part: \(4m + 7m = 11m\).
Are xy and yx like terms?
Yes. The order of multiplying does not matter, so \(xy = yx\).
Are x squared and x like terms?
No. \(x^2\) and \(x\) have different indices, so \(3x^2 + 2x\) stays as it is.
How can I check my answer?
Substitute small numbers into the question and the answer; they must give the same value.