Apples and oranges: collecting like terms
Tonight's idea
A letter stands for a number we do not know yet. \(3x\) means 3 lots of \(x\). Picture \(x\) as a bag of coins. Every bag holds the same number.
Three bags and two more bags make five bags, however many coins are in each. So \(3x + 2x = 5x\). You counted bags, as you would count apples.
Now take three bags and two loose coins. That is not five of anything, because a bag is not a coin. \(3x + 2\) has to stay as it is. Test it with \(x = 10\): \(3x + 2\) is 32, but \(5x\) is 50.
The pieces joined by + or − are called terms. Terms with the same letter are like terms. Plain numbers are like each other. A different letter is a different kind of bag, so \(3x + 2y\) stays put too.
Add or subtract like terms by counting them: \(7y - 4y = 3y\). Unlike terms stay separate. A letter on its own, \(x\), means \(1x\).
Watch one: \(5x + 4 + 2y - 2x + 3 + y\)
Sort the terms into kinds. Each term keeps the sign in front of it.
The \(x\) terms: \(5x - 2x = 3x\). The \(y\) terms: \(2y + y = 3y\). The plain numbers: \(4 + 3 = 7\).
\[5x + 4 + 2y - 2x + 3 + y = 3x + 3y + 7\]Check with any numbers, say \(x = 1\) and \(y = 2\). The long version gives \(5 + 4 + 4 - 2 + 3 + 2 = 16\). The short one gives \(3 + 6 + 7 = 16\). Same value, less writing.
Your turn
- Simplify: (a) \(4x + 5x\); (b) \(7a - 2a + a\); (c) \(6n + 3 + 2n + 8\); (d) \(9y + 4 - 3y - 4\).
- Simplify: (a) \(3x + 2y + 4x + y\); (b) \(5a + 2b - 3a + 6 - 2b\); (c) \(4p - 6p + 5\).
- A rectangle is \(x\) cm wide and \(2x + 3\) cm long. (a) Add the four sides to find the perimeter, written as simply as you can. (b) Find the perimeter when \(x = 4\) in two ways: from your answer to (a), and from the side lengths.
- Sam says \(3x + 2 = 5x\) "because 3 + 2 = 5". (a) Work out \(3x + 2\) and \(5x\) when \(x = 4\). Is Sam right? (b) There is exactly one value of \(x\) for which the two sides are equal. Find it. (c) Why does that one lucky value not make Sam's rule true?
Stretch: a number pyramid
In a number pyramid each brick is the sum of the two bricks under it. The bottom row is \(x\), \(3\), \(2x\). Write the two middle bricks and the top brick as simply as you can. The top brick is 21. Find \(x\).
Answers
- (a) \(9x\). (b) \(6a\). (c) \(8n + 11\). (d) \(6y\): the \(+4\) and \(-4\) cancel.
- (a) \(7x + 3y\). (b) \(2a + 6\): the \(2b\) and \(-2b\) cancel. (c) \(-2p + 5\), also written \(5 - 2p\).
- (a) \(x + (2x + 3) + x + (2x + 3) = 6x + 6\). (b) \(6 \times 4 + 6 = 30\) cm. The sides are 4 and 11, and \(4 + 11 + 4 + 11 = 30\) cm.
- (a) \(3 \times 4 + 2 = 14\) and \(5 \times 4 = 20\). Not equal, so Sam is wrong. (b) \(5x\) is \(3x + 2x\), so the sides match only when \(2x = 2\), which is \(x = 1\). Check: 5 and 5. (c) A rule for simplifying must work for every value of \(x\), not for one.
Stretch. Middle bricks: \(x + 3\) and \(2x + 3\). Top: \(x + 3 + 2x + 3 = 3x + 6\). Then \(3x + 6 = 21\) gives \(3x = 15\), so \(x = 5\). Check: bottom 5, 3, 10; middle 8, 13; top 21.