Mathematics notes, chapter 5
Algebraic expressions
Textbook Mathematics Form 1 (DLP), pages 104–120
What this chapter covers
- Use letters to represent quantities with unknown values, and say whether a variable has a fixed or a varied value, with a reason (5.1)
- Write algebraic expressions for situations (5.1)
- Find the value of an algebraic expression by substituting given values of the variables (5.1)
- Identify the terms in an algebraic expression and state the coefficient of a variable in a term (5.1)
- Identify like terms and unlike terms (5.1)
- Add and subtract two or more algebraic expressions by collecting like terms (5.2)
- Write repeated multiplication in power form, such as a × a × a = a³ and (a + b)(a + b) = (a + b)² (5.2)
- Multiply and divide algebraic expressions with one term (5.2)
A tour company offers a holiday package to Pulau Pangkor: RM380 for an adult and RM280 for a child. How much does a family pay? It depends on how many adults and children go. If we let a be the number of adults and c the number of children, the total cost is 380a + 280c ringgit. That is an algebraic expression. In this chapter you will learn to use letters for unknown values, write and simplify expressions, and work out their values.
The word "algebra" comes from the Arabic word al-jabr, from a book by Muhammad Ibn Musa Al-Khwarizmi, who is called the "Father of Algebra".
| English | Bahasa Melayu |
|---|---|
| variable | pemboleh ubah |
| letter | huruf |
| algebraic expression | ungkapan algebra |
| term, algebraic term | sebutan, sebutan algebra |
| coefficient | pekali |
| like terms, unlike terms | sebutan serupa, sebutan tidak serupa |
5.1Variables and algebraic expressions
Variables: fixed or varied?
Mr Lim gives RM50 to a charity every month, so we know his amount. Encik Azlan gives the same amount every month, but we are not told how much. Madam Kavitha's amount depends on her shop's profit each month. Encik Azlan's and Madam Kavitha's amounts are unknown.
A variable is a quantity with an unknown value. We use a letter to represent it, for example RMx for Encik Azlan and RMy for Madam Kavitha.
| Type | Meaning | Example |
|---|---|---|
| Fixed value | The quantity stays the same all the time. | x, Encik Azlan's monthly donation (the same every month) |
| Varied value | The quantity changes over time. | y, Madam Kavitha's donation (changes with the profit) |
Example 1 (textbook p. 106)
- The annual interest rate for a fixed deposit at a bank. Let k be the interest rate. k has a fixed value because the rate does not change throughout the year.
- The time Faizal takes to travel from his house to school every day. Let t be the time. t has a varied value because the time changes from day to day (traffic, weather).
To answer fully you need three things: the letter, "fixed" or "varied", and a reason starting with "because …".
Choose a letter and say whether each variable is fixed or varied. a) The mass of each student in your class. b) The distance between Arman's house and his school. c) The temperature at the peak of Gunung Kinabalu during a day.
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a) m: varied, because different students have different masses.
b) d: fixed, because the house and the school do not move.
c) T: varied, because the temperature changes during the day.
(Any sensible letter is accepted.)
b) d: fixed, because the house and the school do not move.
c) T: varied, because the temperature changes during the day.
(Any sensible letter is accepted.)
Writing algebraic expressions
A bag holds an unknown number of blocks, x. Each change to the situation changes the expression for the total number of blocks.
An algebraic expression is made of numbers and variables joined by operations, for example x + 3, x − 4, x2, 2x and 2x + y + 2.
| Words | Expression |
|---|---|
| 3 more than x / x increased by 3 | x + 3 |
| subtract 7 from x / 7 less than x | x − 7 |
| 4 times x / twice y | 4x / 2y |
| the sum of y and z divided by 9 | y + z9 |
| m kg at RMp per kg | mp (RM) |
We write 2 × x as 2x (number first, no × sign), and x × y × p as xyp. 1x is written simply as x.
"Subtract 7 from x" means x − 7, not 7 − x. The number you start with comes first.
Example 2 (textbook p. 108)
Yusri buys 5 apples at x sen each and 8 oranges at y sen each. Write an expression for the total amount he pays.
- Apples: 5 × x = 5x
- Oranges: 8 × y = 8y
- Total payment = 5x + 8y sen
Example 3: an area (textbook p. 112)
- Area of the park = x × y = xy m²
- Area of the flower bed = 15 × 15 = 225 m²
- Area covered with bricks = (xy − 225) m²
Write an expression for: a) the number of people in x tents if each tent holds 4 people; b) the sum of y and z, divided by 9; c) Nazmi is h years old, Jagjit is k years old and Izhar is twice as old as Jagjit. The difference between the ages of Nazmi and Izhar (Izhar is older).
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a) 4x
b) y + z9
c) Izhar is 2k years old, so the difference is 2k − h years.
b) y + z9
c) Izhar is 2k years old, so the difference is 2k − h years.
The value of an algebraic expression
To find the value of an expression, substitute (replace) each variable with its value, then calculate. Put each value in brackets.
Example 4 (textbook p. 109)
Given x = 3 and y = 2, find the value of 8x − 5y + 7.
- 8x − 5y + 7 = 8(3) − 5(2) + 7
- = 24 − 10 + 7
- = 21
Example 5 (textbook p. 109)
In a class, 13 of the boys are members of Kadet Remaja Sekolah, and all the girls except 9 are members.
- Let x be the number of boys and y the number of girls. Members = 13x + y − 9
- When x = 12 and y = 16: 13(12) + 16 − 9 = 4 + 16 − 9 = 11 members
With a negative value, keep the brackets. If r = −4, then −4r = −4(−4) = +16, not −16.
On a scientific calculator: type the expression with ALPHA X and ALPHA Y, press CALC, then enter each value and press =.
Given p = 5, q = 2 and r = −4, find the value of: a) 2p + q b) 3q − 4r + 8 c) 5(p − r) d) r2 + 7q − 3
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a) 2(5) + 2 = 10 + 2 = 12
b) 3(2) − 4(−4) + 8 = 6 + 16 + 8 = 30
c) 5(5 − (−4)) = 5(9) = 45
d) −42 + 7(2) − 3 = −2 + 14 − 3 = 9
b) 3(2) − 4(−4) + 8 = 6 + 16 + 8 = 30
c) 5(5 − (−4)) = 5(9) = 45
d) −42 + 7(2) − 3 = −2 + 14 − 3 = 9
Terms and coefficients
An algebraic term is a product of a number and variables, such as 2x or 3xy. The terms of an expression are separated by + and − signs. A number on its own, such as 13, is also a term.
Example 6: identify the terms (textbook p. 110)
| Expression | Terms | Number of terms |
|---|---|---|
| x + 5x | x, 5x | 2 |
| x8 − 7y | x8, 7y | 2 |
| pq − 2q + 13 | pq, 2q, 13 | 3 |
| m2 − 2m + n − 6 | m2, 2m, n, 6 | 4 |
A term can be written as a product of a variable and its factor. That factor is the coefficient of the variable.
| 3xy written as | Coefficient | of the variable |
|---|---|---|
| 3 × xy | 3 | xy |
| 3x × y | 3x | y |
| 3y × x | 3y | x |
Example 7 (textbook p. 111)
In the term −3k2mn, state the coefficient of a) k2mn b) −mn c) 3k2
- −3k2mn = −3 × k2mn, so the coefficient of k2mn is −3.
- −3k2mn = 3k2 × (−mn), so the coefficient of −mn is 3k2.
- −3k2mn = −mn × 3k2, so the coefficient of 3k2 is −mn.
The sign belongs to the term in front of it. In −3k2mn the coefficient of k2mn is −3, not 3. Check: coefficient × variable must give back the whole term.
In the term −8xy2, state the coefficient of a) xy2 b) 8x c) y2 d) −x
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a) −8 (−8 × xy2)
b) −y2 (8x × (−y2))
c) −8x (−8x × y2)
d) 8y2 (−x × 8y2)
b) −y2 (8x × (−y2))
c) −8x (−8x × y2)
d) 8y2 (−x × 8y2)
Like terms and unlike terms
Like terms have the same variables with the same powers. The numbers in front can be different. Unlike terms have different variables, or the same variables with different powers.
| Like terms | Why | Unlike terms | Why |
|---|---|---|---|
| −2m and 37m | both m | 2v and 2w | v and w are different |
| 6ab and 7ba | ba is the same as ab | 5xy and −yz | xy and yz are different |
| 5k2 and −k2 | both k2 | 9x2 and 8x | powers 2 and 1 are different |
Example 8 (textbook pp. 111–112)
- 4xy and xy2: like (same variables xy)
- 12pq and 12pr: unlike (pq and pr are different)
- 3abc and 0.5bca: like (abc is the same as bca)
- −7h and 6h2: unlike (the powers of h are different)
Same numbers do not make like terms: 12pq and 12pr are unlike. Only the variables and their powers matter.
Like or unlike? a) 5k, −0.1k b) 4y, y2 c) 4srt, 11rts d) 3ab2, −8bc
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a) Like b) Unlike (different powers) c) Like (srt = rts) d) Unlike (ab and bc are different)
5.2Algebraic expressions involving basic arithmetic operations
Adding and subtracting algebraic expressions
To add or subtract expressions: remove the brackets, gather the like terms, then add or subtract the like terms. Unlike terms cannot be combined.
| Removing brackets after a − sign | Result |
|---|---|
| −(a + b) | −a − b |
| −(a − b) | −a + b |
| −(−a + b) | +a − b |
| −(−a − b) | +a + b |
A + sign before the brackets: the signs inside stay the same. A − sign before the brackets: every sign inside changes. When you move a term, move its sign with it.
Example 9 (textbook p. 113)
a) (3x + 5y) + (8x − y − 9)
- = 3x + 5y + 8x − y − 9 (remove brackets)
- = 3x + 8x + 5y − y − 9 (gather like terms)
- = 11x + 4y − 9
b) (12mn − 4p) + (6 + 7p) − (10mn + p − 2)
- = 12mn − 4p + 6 + 7p − 10mn − p + 2 (the last bracket: + becomes −, − becomes +)
- = 12mn − 10mn − 4p + 7p − p + 6 + 2
- = 2mn + 2p + 8
In −(10mn + p − 2) many students change only the first sign and write −10mn + p − 2. Every term inside must change: −10mn − p + 2.
Example 10: a rope (textbook p. 117)
- Length of rope = 2d + (d + 2y) + (3y − 2d)
- = 2d + d − 2d + 2y + 3y
- = (d + 5y) cm
Simplify:
a) (3x − 2y) + (5x + 9y)
b) (6ab + 2bc + 10) − (ab + 3bc − 2)
c) (4xy + 5k) − (−3k + 7) + (13xy − k)
a) (3x − 2y) + (5x + 9y)
b) (6ab + 2bc + 10) − (ab + 3bc − 2)
c) (4xy + 5k) − (−3k + 7) + (13xy − k)
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a) 3x + 5x − 2y + 9y = 8x + 7y
b) 6ab − ab + 2bc − 3bc + 10 + 2 = 5ab − bc + 12
c) 4xy + 13xy + 5k + 3k − k − 7 = 17xy + 7k − 7
b) 6ab − ab + 2bc − 3bc + 10 + 2 = 5ab − bc + 12
c) 4xy + 13xy + 5k + 3k − k − 7 = 17xy + 7k − 7
Repeated multiplication
The area of a square and the volume of a cube are repeated multiplications.
a × a × a × … × a (n times) = an, and (a + b) × (a + b) × … × (a + b) (n times) = (a + b)n. The small number n is the power: it counts how many times the expression is multiplied.
Example 11 (textbook p. 115)
- m × m × m × m = m4 (4 times)
- (x + 7) × (x + 7) = (x + 7)2 (2 times)
- (p − 3q) × (p − 3q) × (p − 3q) = (p − 3q)3 (3 times)
- The other way round: (9p − q)3 = (9p − q)(9p − q)(9p − q)
m × m × m × m = m4, not 4m. 4m means m + m + m + m.
Keep the brackets: (x + 7) × (x + 7) = (x + 7)2. Writing x + 72 means x + 49, which is different.
a) Simplify (6a − 1) × (6a − 1).
b) Simplify (8x + 3y) × (8x + 3y) × (8x + 3y).
c) Write (h − 4k)3 as a repeated multiplication.
b) Simplify (8x + 3y) × (8x + 3y) × (8x + 3y).
c) Write (h − 4k)3 as a repeated multiplication.
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a) (6a − 1)2
b) (8x + 3y)3
c) (h − 4k)(h − 4k)(h − 4k)
b) (8x + 3y)3
c) (h − 4k)(h − 4k)(h − 4k)
Multiplying and dividing terms
Multiplying: write each term as a product of factors, then multiply number with number and variable with the same variable. Dividing: write it as a fraction and cancel the common factors.
Example 12: multiplying (textbook p. 116)
Simplify 3ab2 × 4a3b.
- = 3 × a × b × b × 4 × a × a × a × b
- = 3 × 4 × a × a × a × a × b × b × b (gather numbers and the same variables)
- = 12a4b3
Example 13: dividing (textbook p. 116)
Simplify 20m4n2 ÷ 5m2n3.
- = 20m4n25m2n3 (write as a fraction)
- = 20 × m × m × m × m × n × n5 × m × m × n × n × n
- Cancel 5, m × m and n × n: = 4m2n
Example 14: multiply and divide (textbook p. 116)
Simplify 21xy × 6x ÷ 14y3z.
- = 21 × x × y × 6 × x14 × y × y × y × z
- 21 × 6 = 126 and 126 ÷ 14 = 9; one y cancels.
- = 9x2y2z
3x × 5x3 = 15x4. Multiply the numbers (3 × 5 = 15), do not add them (8x4 is wrong). The x's give x × x × x × x = x4.
When dividing, a factor left only at the bottom stays at the bottom: 4ab3 ÷ 6a2b = 2b23a, not 2b23a.
Example 15: a rectangle (textbook p. 117)
A rectangle has an area of 12a3b2 cm² and a width of 3ab cm. Length = 12a3b2 ÷ 3ab = 12 × a × a × a × b × b3 × a × b = 4a2b cm. Check: 3ab × 4a2b = 12a3b2 ✓
Simplify: a) 3x × 5x3 b) −4mn × 7m2 c) 8x6y4 ÷ 2xy3 d) 2mn × 5m2 ÷ 3n3
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a) 3 × 5 × x × x × x × x = 15x4
b) −4 × 7 × m × m × m × n = −28m3n
c) 8x6y42xy3 = 4x5y
d) 2 × 5 × m × m × m × n3 × n × n × n = 10m33n2
b) −4 × 7 × m × m × m × n = −28m3n
c) 8x6y42xy3 = 4x5y
d) 2 × 5 × m × m × m × n3 × n × n × n = 10m33n2
Summary
| Idea | Key fact | Example |
|---|---|---|
| Variable | A letter for an unknown quantity: fixed value (always the same) or varied value (changes over time) | days in a week: fixed; daily rainfall: varied |
| Expression | Numbers and variables joined by operations | 5x + 8y, x2, xy − 225 |
| Value | Substitute in brackets, then calculate | x = 3, y = 2: 8(3) − 5(2) + 7 = 21 |
| Coefficient | The factor multiplying a variable, sign included | In −3k2mn, coefficient of k2mn is −3 |
| Like terms | Same variables, same powers | 2x and −35x; 6ab and 7ba |
| Unlike terms | Different variables or different powers | 2x and 6y; 2x and −7x2 |
| Add, subtract | Remove brackets (− changes every sign), gather like terms | 5a + 6a = 11a; 12bc − 3bc = 9bc |
| Power | Repeated multiplication | m × m × m = m3; (a + b)(a + b) = (a + b)2 |
| Multiply, divide | Write out the factors; multiply or cancel | 5mn × 2n = 10mn2; 36pq ÷ 4p = 9q |