Algebra | Equations and relationships
Topic 2.1 Expressions and substitution
Algebraic expressions
An algebraic expression is made up of terms that contain variables, coefficients and constants.
Learn
1 For each expression, identify the term(s), variable(s), coefficient(s) and the constant (if applicable).
Constant b + 8 7 + a 4v − 9
8n + 2
3a + 4b 9p − 7q 10a + b 12x 5y
5d − 6h − 1
6n − 7m + 5p
3c − 5b + 4
8g + 9h − 10 + i
DRAFT
2 If we have a number represented by the variable n, give an expression for:
a Three added to the number:
c The number multiplied by seven:
e The number subtracted from eleven:
g Nine more than the number:
i The square root of the number:
k The sum of the number and 11:
b Six less than the number:
d The number divided by five:
f Twelve divided by the number:
h The number multiplied by itself:
j The difference between the number and 7:
l The product of the number and 4:
Multiplying and dividing terms
Any two terms can be multiplied together. We usually remove multiplication signs when working with variables, and often choose to write variables in alphabetical order (which we can do because multiplication is commutative). Doing this usually makes terms easier to read and compare. As with numbers, exponents show repeated multiplication.
8 × a = 8a 7 × 3b = 7 × 3 × b = 21b c × c = c 2
Any term can be divided by any non-zero term. Division symbols are usually replaced by fractions, where the term before the division symbol is the numerator and the term after is the denominator. This helps to avoid ambiguity in the order of operations.
x ÷ 3 = x 3 (b + 3) ÷ 4 = b + 3 4
Learn
1 Simplify these expressions. a 3 × w =
7 × b =
10 × y =
5 × z =
2 In algebraic expressions, the multiplication sign (×) is usually removed (e.g. 2 × a = 2a). Explain why the multiplication sign must not be removed in an expression that contains only digits e.g. (2 × 7).
3 Write these division expressions in fraction form.
2a ÷ 7 =
4 Simplify the following.
4 × b =
5 Write a simplified formula for the area of each shape.
DRAFT
Like and unlike terms
Like terms have exactly the same variables (letters).
Are 4a and 4ab like terms?
1 Are the following terms like or unlike? If they are like terms, give two more examples. Terms Like Unlike Examples
Like Unlike
2h, 5h ✓ 3h, h
6g, 4h ✓
7f, 3f
g, 3h
7h, 9h
4h, h
9k, 3k
w, 7w
9p, 7q 2, 6
2g, h
2 Circle the like terms in each list.
DRAFT
3 Explain why ab and ba are like terms.
Adding and subtracting terms
When simplifying expressions with more than one variable, identify like terms and combine them by reading from left to right. Remember, a term includes the sign to the left.
Explore
1 Simplify these expressions. It may help to use the number line provided above. a 6a + a =
5b − b =
3v + 2v = d 6h − 2h =
5a + a =
6b − b = g 7a + 3a =
6h − 2h =
2 Simplify these expressions by adding and subtracting like terms. a 4m + 3 − m =
3n + 7n − 7 =
3 Simplify these expressions.
3
4y + 9y − 2 =
10b − 5b + 7 =
9b − 5b + 8 =
7p + 2p − 7 =
= i 9k + 4m + 7 − 2k
4 Write a simplified formula for the perimeter of each shape.
Substitution
We can substitute (replace) the variables in an algebraic expression or formula with numbers.
Expression Values of variables
Substitution
a + 3 a = 5 5 + 3 = 8
2b − 4 b = 7 2(7) − 4 = 2 × 7 − 4 = 14 − 4 = 10
5f + 6g f = 3, g = 2 5(3) + 6(2) = 5 × 3 + 6 × 2 = 15 + 12 = 27
A = b × h b = 6, h = 5 A = 6 × 5 = 30
Deepen
1 Find the value of the expression by substituting the following numbers for the variables: a = 2, b = 5, c = 8, d = 3
a b + 6 = = b 7a = = c 5c − 3 = =
d 4b + 2a = = e d 2 = = f ad = =
g c 2 = = h ac 4 = = i 3 × 4 d = =
2 Find the value of the expressions below by substituting a = 3, b = 15, c = 6 and d = 12
a d ÷ 3 = b ab = c ac ÷ 2 =
d 2b − 3c + 7 = e 2c − d = f 4a + c − 5 =
3 Substitute the numbers into the formulae.
a A = b × h (b = 5 and h = 8) = =
c Cost = $20 + $30h (h = 7) = =
b P = 2b + 2h (b = 3 and h = 5) = =
d F = ma (m = 5 and a = 10) = =
e d = vt (v = 30 and t = 7) = = f Cost = $5 + $6k (k = 9)
4 A taxi charges a flag fall fee of $4 (fee to arrive at your location) and $3 per km If the taxi travels k kilometres:
a Write an expression/formula for the cost of the taxi:
b Use this expression/formula to find the cost of travelling:
i 10 km:
ii 15 km:
iii 50 km
