Unit 11 Section 1 : Equations, Formulae and Identities

In this section we discuss the difference between equations, formulae and identities, and then go on to make use of them.

An equation contains unknown quantities; for example,

3x + 2 = 11

This equation can be solved to determine x.

A formula links one quantity to one or more other quantities; for example,

A = πr2

This formula can be used to determine A for any given value of r.

An identity is something that is always true for any values of the variables that are involved; for example,

2(x + y) ≡ 2x + 2y
and
(x + y)^2 ≡ x^2 + 2xy + y^2

If any pair of values of x and y are substituted, then the left hand side of an identity will generate the same value as the right hand side of that identity.

Example 1

The formula C = (F – 32) is used to convert temperatures in degrees Fahrenheit to degrees Celsius.

(a)

If F = 41, calculate C.

C = × (41 – 32)
C = × 9
C = 5

(b)

If F = 131, calculate C.

C = × (131 – 32)
C = × 99
C = 55

Example 2

A formula states that v = u + at.
Calculate v if u = 10, a = 6.2 and t = 20.

When substituting into equations, you need to be aware that the BODMAS rule applies automatically.

v = u + at
v = 10 + 6.2 × 20
v = 10 + 124
v = 134

Example 3

Solve the following equations:

(a)

7x = 21

7x = 21
x = Dividing both sides by 7
x = 3

(b)

x – 5 = 12

x – 5 = 12
x = 12 + 5Adding 5 to both sides
x = 17

(c)

2x + 1 = 6

2x + 1 = 6
2x = 6 – 1Subtracting 1 from both sides
2x = 5
x = Dividing both sides by 2
x = 2

(d)

5x – 8 = 22

5x – 8 = 22
5x = 22 + 8Adding 8 to both sides
5x = 30
x = Dividing both sides by 5
x = 6

Example 4

One of the following statements is not an identity. Which one?

A ≡ +
Bx - y ≡ y - x
Cx^2 + y^2 ≡ (x + y)^2 - 2xy

An identity will be true for any pair of values x and y. We could test each statement with x = 5 and y = 10 .

Left-hand-side of A = = = = 7.5
Right-hand-side of A = + = + = 2.5 + 5 = 7.5

Therefore LHS of A = RHS of A if x = 5 and y = 10.

LHS of B = x - y = 5 - 10 = -5
RHS of B = y - x = 10 - 5 = 5

Therefore LHS of B ≠ RHS of B if x = 5 and y = 10.

LHS of C = x^2 + y^2 = 5^2 + 10^2 = 25 + 100 = 125
RHS of C = (x + y)^2 - 2xy = (5 + 10)^2 - 2 × 5 × 10 = 15^2 - 100
= 225 - 100 = 125

Therefore LHS of C = RHS of C if x = 5 and y = 10.

So statement B is not an identity. We have not proved that A and C are identities, but we know that they are true for certain values of x and y.

Exercises

Question 1

Solve the following equations:

(a)4x = 12 x =
(b)x - 5 = 8 x =
(c)x + 3 = 9 x =
(d)15 + x = 20 x =
(e)x - 3 = 9 x =
(f) = 7 x =
(g)x + 7 = 22 x =
(h)5x = 30 x =
(i) = 9 x =
Question 2

Solve the following equations:

(a)2x + 1 = 11 x =
(b)4x - 3 = 21 x =
(c)5x - 6 = 4 x =
(d)5x - 9 = 26 x =
(e)9x + 21 = 102 x =
(f)10x - 5 = 35 x =
(g) + 3 = 4 x =
(h) - 9 = 2 x =
(i) + 3 = 11 x =
(j)x + 6 = 5 x =
(k)4x + 20 = 10 x =
(l) + 9 = 4 x =
Question 3

In this question, use the formula,

C = (F – 32)

Calculate C if:

(a)

F = 77

C =
(b)

F = 68

C =
(c)

F = 158

C =
Question 4

In this question, use the formula,

v = u + at

Calculate v if:

(a)

u = 2, a = 3 and t = 7

v =
(b)

u = 3.2, a = 0.8 and t = 5

v =
(c)

u = 30, a = 4 and t = 22

v =
(d)

u = 3.6, a = –0.2 and t = 40

v =
Question 5

The formula for the area of a trapezium is,

A = (a + b)h

Calculate A if:

(a)

a = 4, b = 10 and h = 6

A =
(b)

a = 2, b = 10 and h = 13

A =
(c)

a = 3.2, b = 2.8 and h = 3.2

A =
(d)

a = 4, b = 2.5 and h = 7.2

A =
Question 6

If a = 6, b = 7.5 and c = –2, calculate:

(a)a + b + c x =
(b)ab + c x =
(c)2a + 3b x =
(d)a + 2b + 3c x =
(e)ac x =
(f)a^2 + b^2 x =
(g)a^2 + c^2 x =
(h)ab + bc x =
(i)a(b - c) x =
Question 7

A formula states:

y = 4x - 5

(a)

Calculate y if x = 3.

y =
(b)

Determine x if y = 23.

x =
(c)

Determine x if y = 8.

x =
Question 8

The mean of three numbers is calculated using the formula,

m =

(a)

Calculate m if x = 8, y = 17 and z = 2.

m =
(b)

Determine x if m = 5, y = 6 and z = 7.

x =
(c)

Determine z if m = 18, x = 19 and y = 20.

z =
Question 9

Use the formula C = (F – 32) to determine F when:

(a)

C = 100

F =
(b)

C = 60

F =
(c)

C = 0

F =
Question 10

Which of the following statements are not identities?

A  ≡ 
Bx × y  ≡  y × x
C(x - y)^2  ≡  (y - x)^2
D(a + b)^2  ≡  (a - b)^2
E2xy  ≡  (x + y)^2 - x^2 - y^2
Question 11

Jenny is holding a row of cubes.

You cannot see exactly how many cubes she is holding.

Call the number of cubes she is holding n.

(a)

She joins on two more cubes.

Write an expression for the total number of cubes she is holding now.

(b)

Jenny starts again with n cubes.

One cube is removed.

Write an expression for the number of cubes she is holding now.

(c)

Jenny starts again with n cubes.

Another row of the same length is joined on.

Write an expression for the total number of cubes she is holding now.

(d)

Jacob also has some cubes in his hands.

In one hand there are 2n – 1.
In the other hand there are 2(n – 1) cubes.

Is Jacob holding the same number of cubes in each hand?

2n – 1 = n + (n – 1), so he has one more cube in the hand with 2n – 1 than in the hand with 2(n – 1).
Question 12

(a)

Elin has a bag of marbles.
You cannot see how many marbles are inside the bag.
Call the number of marbles which Elin starts with in her bag n.
Elin puts 5 more marbles into her bag.
Write an expression to show the total number of marbles in Elin's bag now.

(b)

Ravi has another bag of marbles.
Call the number of marbles which Ravi starts with in his bag t.
Ravi takes 2 marbles out of his bag.
Write an expression to show the total number of marbles in Ravi's bag now.

(c)

Jill has 3 bags of marbles.
Each bag has p marbles inside.
Jill takes some marbles out.
Now the total number of marbles in Jill's 3 bags is 3p – 6.
Some of the statements below could be true.
Write down the letter of each statement which could be true.

AJill took 2 marbles out of one of the bags, and none out of the other bags.
BJill took 2 marbles out of each of the bags.
CJill took 3 marbles out of one of the bags, and none out of the other bags.
DJill took 3 marbles out of each of two of the bags, and none out of the other bag.
EJill took 6 marbles out of one of the bags and none out of the other bags.
FJill took 6 marbles out of each of two of the bags, and none out of the other bag.
Question 13

In these walls each brick is made by adding the two bricks underneath it.

(a)

Write an expression for the top brick in this wall.
Write your expression as simply as possible.

(b)

Fill in the missing expressions in the walls shown below.
Write your expressions as simply as possible.

(c)

In this wall, h, j and k can be any whole numbers.
Complete the sentence below.

The top brick of the wall must always be an number.