Algebra · Algebraic Expressions · Lesson 2

Adding and Subtracting Algebraic Expressions

A long expression such as $3 x + 2 y + 2 x + y$ is like a market basket with yams and bags of gari thrown in anyhow. In this lesson you will learn to sort it out — collect the like terms together, add or subtract their coefficients (whole numbers, fractions or decimals) and write the answer in its simplest form. You will also use this to write the perimeter of shapes.

NaCCA B7.2.2.1.2 ⏱ about 45 min

By the end you will be able to

✔Identify like terms and unlike terms in an algebraic expression.
✔Add and subtract like terms by adding or subtracting their coefficients.
✔Simplify expressions with whole-number, fraction and decimal coefficients.
✔Write a simplified expression for the perimeter of a shape.

Key words

Like terms: Terms with exactly the same letter part, e.g. $3 x$ and $- 5 x$ , or $2 ab$ and $7 ab$ .
Unlike terms: Terms whose letter parts are different, e.g. $3 x$ and $3 y$ , or $2 a$ and $2 a^{2}$ .
Coefficient: The number in front of the letter part; in $\frac{3}{4} m$ it is $\frac{3}{4}$ .
Simplify: Collect like terms and write the expression with as few terms as possible.
Rational coefficient: A coefficient that is a whole number, fraction or decimal, e.g. 5, $\frac{1}{2}$ or 0.3.

Curriculum reference

Content standard: B7.2.2.1 Simplify algebraic expressions involving the four basic operations and substituting values to evaluate algebraic expressions.

Indicator: Perform addition and subtraction of algebraic expressions with rational coefficients.

Starter

Yams and gari in one basket

On Monday Mama Esi buys 3 tubers of yam and 2 bags of gari for her chop bar in Kumasi. On Thursday she buys 2 more tubers and 1 more bag. How much did she buy altogether? Nobody says "8 yam-garis"! She has 5 tubers of yam and 3 bags of gari. If one tuber costs GH₵y and one bag costs GH₵g, her spending is $3 y + 2 g + 2 y + g = 5 y + 3 g$ . Algebra keeps yams with yams and gari with gari — that is the whole secret of this lesson.

1. Like terms and unlike terms

A term is a part of an expression separated by + or − signs. Two terms are like terms when their letter parts are exactly the same. Only the coefficient (the number in front) may be different. Like terms are things of the same kind, so they can be counted together.

Terms	Like or unlike?	Reason
$4 x$ and $9 x$	like	both have letter part $x$
$3 ab$ and $- 5 ba$	like	$ba$ is the same as $ab$ ( $b \times a = a \times b$ )
$\frac{1}{2} m$ and $0.3 m$	like	both have letter part $m$
$2 a$ and $2 b$	unlike	different letters
$5 x$ and $5 x^{2}$	unlike	$x$ and $x^{2}$ are different
$7$ and $7 p$	unlike	7 is a constant; $7 p$ has a letter

2. Adding like terms

Think of $x$ as one tuber of yam. Then $x + x + x + x$ is four tubers, which we write $4 x$ . In the same way $4 x + 3 x$ is 4 tubers plus 3 tubers, which is 7 tubers: $7 x$ . To add like terms, add the coefficients and keep the letter part the same.

$x + x = 2 x$
$x + x + x + x + x + x + x + x = 8 x$
$4 x + 3 x + x = 8 x$ (remember $x$ means $1 x$ )
$5 a + 4 a + 2 a + 3 a = 14 a$

When an expression has more than one kind of term, collect each kind together first. We are allowed to change the order of terms in an addition, as long as each term keeps its own sign. Step through the algebra tiles below.

▶ Moving tiles around does not change how many there are. Sorting like with like gives 5x + 3y.

3. Subtracting like terms

Subtracting works the same way: subtract the coefficients and keep the letter part. If you have 5 tubers of yam and sell 2, you have 3 left: $5 x - 2 x = 3 x$ .

▶ Taking away 2 tiles from 5 tiles leaves 3 tiles: only the coefficients change.

In a longer expression, the sign in front of a term belongs to that term and moves with it. In $5 a + 4 b - 9 a + 3 b + 2 a - 7 b$ the a-terms are $+ 5 a$ , $- 9 a$ and $+ 2 a$ ; the b-terms are $+ 4 b$ , $+ 3 b$ and $- 7 b$ .

5a + 4b − 9a + 3b + 2a − 7b = −2a
Kind of term	Terms collected	Coefficients	Result
a-terms	$5 a - 9 a + 2 a$	5 − 9 + 2 = −2	$- 2 a$
b-terms	$4 b + 3 b - 7 b$	4 + 3 − 7 = 0	$0 b = 0$

When a whole bracket is subtracted, the minus sign acts on every term inside it: $- (x + 1) = - x - 1$ . For example, $(3 x + 2) - (x + 1) = 3 x + 2 - x - 1 = 2 x + 1$ .

4. Fractions, decimals and perimeters

Coefficients do not have to be whole numbers. The rule is the same — add or subtract the coefficients — but now you use your fraction and decimal skills.

$\frac{1}{2} x + \frac{1}{4} x = \frac{2}{4} x + \frac{1}{4} x = \frac{3}{4} x$
$\frac{2}{3} a - \frac{1}{6} a = \frac{4}{6} a - \frac{1}{6} a = \frac{3}{6} a = \frac{1}{2} a$
$0.5 m + 1.2 m - 0.3 m = 1.4 m$

The perimeter of a shape is the sum of the lengths of all its sides. When the sides are expressions, add them and collect like terms. A triangle with sides $2 x$ , $3 x$ and $4 y$ has perimeter $2 x + 3 x + 4 y = 5 x + 4 y$ . A rectangle $(3 x + 2)$ m long and $(x + 1)$ m wide has perimeter $(3 x + 2) + (x + 1) + (3 x + 2) + (x + 1) = 8 x + 6$ metres.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Counting like terms

Simplify (a)

x + x + x + x + x + x + x + x

(b)

4 x + 3 x + x

Show the solution

Step 1(a) There are 8 $x$ 's, so the sum is $8 x$ .
Step 2(b) Add the coefficients: 4 + 3 + 1 = 8 (the last $x$ is $1 x$ ).
Step 3So $4 x + 3 x + x = 8 x$ .

Answer: (a) $8 x$ (b) $8 x$

Example 2

Two kinds of term

Simplify

5 a + 4 b + 2 a + 3 b

Show the solution

Step 1Collect the a-terms: $5 a + 2 a = 7 a$ .
Step 2Collect the b-terms: $4 b + 3 b = 7 b$ .
Step 3 $7 a$ and $7 b$ are unlike, so stop: $7 a + 7 b$ .

Answer: $7 a + 7 b$

Example 3

Adding and subtracting together

Simplify

5 a + 4 b - 9 a + 3 b + 2 a - 7 b

Show the solution

Step 1a-terms: $5 a - 9 a + 2 a = (5 - 9 + 2) a = - 2 a$ .
Step 2b-terms: $4 b + 3 b - 7 b = (4 + 3 - 7) b = 0 b = 0$ .
Step 3Answer: $- 2 a$ .
Step 4Check with $a = 1, b = 1$ : 5 + 4 − 9 + 3 + 2 − 7 = −2 ✓

Answer: $- 2 a$

Example 4

Fraction coefficients

Simplify

\frac{1}{2} x + \frac{2}{3} x

Show the solution

Step 1Add the coefficients: $\frac{1}{2} + \frac{2}{3}$ .
Step 2Common denominator 6: $\frac{3}{6} + \frac{4}{6} = \frac{7}{6}$ .
Step 3So the sum is $\frac{7}{6} x$ (or $1 \frac{1}{6} x$ ).

Answer: $\frac{7}{6} x$

Example 5

Perimeter of a rectangle

A rectangular plot is

(3 x + 2)

m long and

(x + 1)

m wide. Write a simplified expression for its perimeter.

Show the solution

Step 1Perimeter = length + width + length + width.
Step 2 $= (3 x + 2) + (x + 1) + (3 x + 2) + (x + 1)$ .
Step 3x-terms: $3 x + x + 3 x + x = 8 x$ . Numbers: $2 + 1 + 2 + 1 = 6$ .
Step 4Perimeter $= 8 x + 6$ metres.

Answer: $(8 x + 6)$ m

Summary

Like terms have exactly the same letter part ( $3 ab$ and $5 ba$ are like); unlike terms cannot be combined.
To add or subtract like terms, add or subtract the coefficients and keep the letter part.
The sign in front of a term belongs to it; collect each kind of term with its sign.
$2 a + 3 b$ is already simplest — it is not $5 ab$ .
Fraction and decimal coefficients follow the same rule; perimeter = sum of all the sides, simplified.

Yams and gari in one basket

1. Like terms and unlike terms

2. Adding like terms

3. Subtracting like terms

4. Fractions, decimals and perimeters

Worked examples

Counting like terms

Two kinds of term

Adding and subtracting together

Fraction coefficients

Perimeter of a rectangle

Summary

Practice: 15 questions