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Like terms

Algebra · level 4, lesson 3 of 7

A question to start

Look at this long expression:

\[x + 4 + x + x - 3 + 2x + 6 - x + 1\]

Put \(x = 10\) into it and work it all out. Then put \(x = 10\) into the short expression \(4x + 8\). You get 48 both times. Try \(x = 3\): both give 20. Try \(x = 0\): both give 8.

The long expression and the short one are the same expression, written in two ways. Turning the long one into the short one is called simplifying, and it rests on one idea: you can only add things of the same kind. This lesson shows that idea with tiles, and explains why \(x\) and \(x^2\) can never be added together.

The idea

Terms and like terms

Remember that the terms of an expression are the pieces joined by \(+\) and \(-\). In \(3x + 5 + 2x - 1\) the terms are \(3x\), \(5\), \(2x\) and \(-1\). The sign in front of a term belongs to it, like a label stuck to its front.

Like terms have exactly the same letter part, with the same powers. They may have different coefficients.

Like terms Not like terms
\(3x\) and \(2x\) \(3x\) and \(3\)
\(5\) and \(-1\) (both constants) \(x\) and \(x^2\)
\(4ab\) and \(-ab\) \(ab\) and \(a\)
\(7y^2\) and \(y^2\) \(y^2\) and \(2y\)

Collecting like terms means adding or subtracting the like terms, to make the expression shorter.

Tiles: a picture of each kind

Algebra tiles are a way to see terms. Each kind of term gets its own shape:

  • a small square, 1 by 1, stands for the number 1;
  • a long bar, 1 by \(x\), stands for \(x\) (its length \(x\) is unknown, so we just draw it long);
  • a big square, \(x\) by \(x\), stands for \(x^2\) (its area is \(x \times x\)).
x² x by x x 1 by x 1 1 by 1 three different kinds of tile: x², x and 1

To simplify, you sort the tiles into piles of the same kind and count each pile. Here is \(2x + 3 + x + 1\):

2x + 3 + x + 1 = 3x + 4

Three bars and four small squares: \(2x + 3 + x + 1 = 3x + 4\).

Why \(x\) and \(x^2\) are not like terms

Look at the tiles again. A bar has area \(x\); a big square has area \(x^2\). You cannot put a bar and a big square together and call the result "two bars" or "two big squares", because they are different shapes. So \(x + x^2\) stays as it is: it does not simplify to \(2x^2\) or to \(2x\) or to \(x^3\).

You can test this with a number. Take \(x = 3\):

  • \(x + x^2 = 3 + 9 = 12\);
  • \(2x^2 = 2 \times 9 = 18\);
  • \(2x = 6\), and \(x^3 = 27\).

None of them is 12, so none of them is the same as \(x + x^2\). The same is true of \(3x\) and \(3\): a bar and a small square are different sizes (unless \(x\) happens to be 1), so \(3x + 3\) is not \(6x\) and not \(6\).

Why collecting like terms works

\(3x + 2x\) means "3 lots of \(x\), then 2 more lots of \(x\)". That is 5 lots of \(x\), whatever \(x\) is: \(3x + 2x = 5x\). It is exactly like 3 apples plus 2 apples making 5 apples. The letter is the "kind of thing", and the coefficient says how many. Adding like terms adds the coefficients and keeps the letter part the same.

Subtracting works the same way: \(7x - 4x\) is 7 lots of \(x\), take away 4 lots, leaving \(3x\).

Moving terms around

You may move the terms of an expression into any order you like, as long as each term keeps its own sign. That is because adding can be done in any order, and a subtraction is just adding a negative: \(5 - 1\) is \(5 + (-1)\).

\[3x + 5 + 2x - 1 = 3x + 2x + 5 - 1 = 5x + 4\]

A good method:

  1. Underline (or colour) each kind of term in a different way, including the sign in front.
  2. Collect each kind.
  3. Write the answer, usually with the highest power first and the constant last.
  4. Check by substituting a number into the long and short versions.

Worked examples

1. The outline example. Simplify \(3x + 5 + 2x - 1\).

  • \(x\) terms: \(3x + 2x = 5x\). Constants: \(5 - 1 = 4\).
  • Answer: \(5x + 4\).
  • Check with \(x = 2\): long: \(6 + 5 + 4 - 1 = 14\); short: \(10 + 4 = 14\). ✓

2. Negatives. Simplify \(4a - 7 - 6a + 2\).

  • \(a\) terms: \(4a - 6a = -2a\). Constants: \(-7 + 2 = -5\).
  • Answer: \(-2a - 5\).
  • Check with \(a = 1\): long: \(4 - 7 - 6 + 2 = -7\); short: \(-2 - 5 = -7\). ✓

3. Two letters. Simplify \(7a - 3b + 2a + 5b\).

  • \(a\) terms: \(7a + 2a = 9a\). \(b\) terms: \(-3b + 5b = 2b\).
  • Answer: \(9a + 2b\).

4. Squares. Simplify \(x^2 + 3x + 2x^2 - x + 4\).

  • \(x^2\) terms: \(x^2 + 2x^2 = 3x^2\). \(x\) terms: \(3x - x = 2x\). Constant: 4.
  • Answer: \(3x^2 + 2x + 4\).
  • Check with \(x = 2\): long: \(4 + 6 + 8 - 2 + 4 = 20\); short: \(12 + 4 + 4 = 20\). ✓

5. A perimeter. A triangle has sides \(x\), \(x + 2\) and \(2x - 1\) cm. Write its perimeter as simply as possible.

  • \(x + (x + 2) + (2x - 1) = x + x + 2x + 2 - 1 = 4x + 1\) cm.
  • If \(x = 5\) the sides are 5, 7 and 9, and the perimeter is 21; \(4 \times 5 + 1 = 21\). ✓

6. The starting puzzle. \(x + 4 + x + x - 3 + 2x + 6 - x + 1\): the \(x\) terms are \(1 + 1 + 1 + 2 - 1 = 4\) lots of \(x\); the constants are \(4 - 3 + 6 + 1 = 8\). So it is \(4x + 8\).

Try it

  1. Simplify \(6y + 2y\).
  2. Simplify \(5n + 3 + 2n + 8\).
  3. Simplify \(9k - 4 - 3k + 10\).
  4. Simplify \(2p - 5q + 4p + 7q\).
  5. Simplify \(3x^2 + 2x - x^2 + 5x - 6\).
  6. A rectangle is \(3x + 1\) long and \(x + 2\) wide. Write its perimeter as simply as possible.
  7. Leah writes \(4x + 3 = 7x\). Use tiles or a number to explain why she is wrong.
  8. Which of these are equal to \(2x + 6\) for every \(x\)? (a) \(x + 6 + x\); (b) \(8x\); (c) \(3x + 2 - x + 4\); (d) \(2(x + 3)\).
Answers
  1. \(8y\).
  2. \(7n + 11\).
  3. \(6k + 6\).
  4. \(6p + 2q\).
  5. \(2x^2 + 7x - 6\).
  6. \(2(3x + 1) + 2(x + 2) = 6x + 2 + 2x + 4 = 8x + 6\). (Or add the four sides: \(3x + 1 + x + 2 + 3x + 1 + x + 2 = 8x + 6\).)
  7. \(4x\) is four bars and 3 is three small squares: different kinds of tile, so they cannot make seven bars. With \(x = 2\): \(4 \times 2 + 3 = 11\), but \(7 \times 2 = 14\).
  8. (a) yes: \(x + x = 2x\). (b) no: with \(x = 2\), \(2x + 6 = 10\) but \(8x = 16\). (They happen to agree when \(x = 1\), both 8, but one match is not enough: they must agree for every \(x\).) (c) yes: \(3x - x = 2x\) and \(2 + 4 = 6\). (d) yes: two lots of \(x + 3\) is two bars and six small squares.

Common mistake

Collecting unlike terms. \(5x + 2\) is not \(7x\), and \(x + x^2\) is not \(2x^2\). If the letter parts are not exactly the same, the terms stay separate.

Losing a minus sign. In \(8 - 3a + 5a\), the term is \(-3a\), so the \(a\) terms give \(-3a + 5a = 2a\), and the answer is \(8 + 2a\). Keep each sign glued to the term after it when you move terms around.

Where it leads