Counting copies of the same thing
3a means 3 lots of a, and 2a means 2 lots of a. Put them together and there are 5 lots of a: 3a + 2a = 5a. It works just like 3 apples and 2 apples making 5 apples.
The number in front of a letter is called its coefficient. It counts how many copies of the letter part there are. Adding 3a and 2a just adds the counts, 3 + 2 = 5. The letter part, a, stays as it is.
Every piece is the same length, a. The 3 colored pieces and the 2 white pieces lie end to end as one bar of 5 pieces: 3a + 2a = 5a.
3x + 2x = 5x: every x strip is the same length, so the two groups lie end to end as one bar of 5; the 2y bar is a different length and stays 2y
Make the x strips collect into 8x
Drag the handles to change how many x strips are in each group. Every x strip is the same length, so all of them join into one bar. The y strips are a different length and stay 2y. Make the x strips collect into 8x.
Like terms and unlike terms
Terms with exactly the same letter part are called like terms. 3a and 2a are like terms, because both are lots of a. Drawn as rectangles, 3a is 3 by a and 2a is 2 by a. Both have a side of length a, so they fit together into one rectangle, 5 by a, whose area is 5a.
A 2b is a rectangle 2 by b. Its side b is a different length from a, so it cannot join the others. 3a and 2b are unlike terms, and 3a + 2b cannot be written any more simply.
The letter part includes its power. means x × x, which is a different quantity from x, just as an area is different from a length. So 3x and are unlike terms, and cannot be simplified.
3a and 2a share the side a, so together they make a rectangle 5 by a: 5a.
Collecting like terms
Simplify 4a + 7b − a + 2b. Each sign belongs to the term just after it, so the terms are 4a, +7b, −a and +2b. The term −a means take away one lot of a.
Collect the a terms: 4a − a = 3a. Collect the b terms: 7b + 2b = 9b. So 4a + 7b − a + 2b = 3a + 9b. The a terms and the b terms are unlike, so this is as simple as it gets.
Check with numbers. When a = 2 and b = 1, the original expression is 8 + 7 − 2 + 2 = 15, and 3a + 9b is 6 + 9 = 15.
The usual mistakes
Multiplying the coefficients. 3a + 2a is 5a, not 6a: the terms are added, so only the counts add.
Changing the letter part. 3a + 2a is not . Adding copies of a never makes a × a.
Joining unlike terms. 3a + 2b is not 5ab, because a and b stand for different numbers.
Worked example: The Fence Round a Plot Whose Sides Are Given in x and y
Question A plot of land has four sides. In meters they are 3x + 2y, x + 4y, 2x − y and x + 4y. (a) Write the length of the fence round the plot as a simplified expression. (b) Find the length of the fence when x = 5 and y = 3.
1.The length of the fence is the sum of the four sides: (3x + 2y) + (x + 4y) + (2x − y) + (x + 4y).
The fence is the perimeter: (3x + 2y) + (x + 4y) + (2x − y) + (x + 4y). 2.Collect the x terms: 3x + x + 2x + x = 7x.
Collect the x terms: 3x + x + 2x + x = 7x. 3.Collect the y terms, keeping the minus sign with the y it belongs to: 2y + 4y − y + 4y = 9y.
Collect the y terms, keeping the minus sign with its y: 2y + 4y − y + 4y = 9y. 4.(a) The length of the fence is (7x + 9y) m. 7x and 9y are unlike terms, so the expression cannot be made any simpler.
(a) The fence is (7x + 9y) m long. 7x and 9y are unlike terms. 5.Substitute x = 5 and y = 3: 7 × 5 + 9 × 3 = 35 + 27 = 62.
Substitute x = 5 and y = 3: 7 × 5 + 9 × 3 = 35 + 27. 6.(b) The fence is 62 m long. Check: the four sides are 21 m, 17 m, 7 m and 17 m, and 21 + 17 + 7 + 17 = 62.
(b) The fence is 62 m long, and the four sides are 21 + 17 + 7 + 17 = 62 m.
Answer: (a) (7x + 9y) m; (b) 62 m
Common mistakes
- Adding the x terms to the y terms to get 16xy. x and y stand for different lengths, so 7x and 9y are unlike terms and stay separate.
- Counting −y as +y and writing 11y. The third side is 2x − y, so its y term is subtracted: 2y + 4y − y + 4y = 9y.