Formulae - StudentZone

Formulae
An algebraic expression describes a mathematical relationship or pattern.
What is an equation?
An equation involves some algebraic expressions.
An equation has an equals sign. This means you can find solutions and find
values for the unknown variables.
2x + 3 = 7, y = 5x + 4.
What is a formula?
These are equations that relate to some specific relationship, rather than
being an abstract equation.
e.g. A = ½ bh, C = 2Õr, etc.
If the equation ‘equals’ something then it must be balanced around the equals
sign. To solve the equation the balance MUST be maintained.
1. I think of a number, double it, add 3 and the answer is 131. What is my
number?
· Write an equation of the puzzle
· Draw a picture of the process
· Work backwards and ‘undo’ the puzzle and write the inverse
equation.
2. I think of another number, subtract 1 and multiply by 3, then subtract 7
and the answer is 8. What is my number?
· What is the equation of the puzzle?
· Write the inverse equation.
3. Find the value of m in the equation: 6m – 27 = 0
4. The formula for the area of a square is x²
· Find the length of the side of a square whose area is 25cm²
· Why is the formula squared?
5. To convert miles into kilometres you multiply by 1.609
· Write this as a formula
· Substitute some figures to test the formula
(Write what you did in your own words. This will help if the process
is unclear.)
If an unknown value is ‘tied up’ with other numbers and values, i.e. it’s not
alone on one side of the equals sign, you may need to rearrange a formula.
This is called changing the subject of the formula. The balance must be
maintained.
Prepared with reference to and examples from:
Edexcel GCSE Mathmatics­ Int.(2001) Heinemann
Haylock,D(2001) Mathmatics Explained for Primary Teachers London: Paul Chapman Publishing
Suggate,J et al(2001) Mathmatical Knowledge for Primary Teachers London:David Fulton Publishers
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Formulae
6. Rearrange the following equations:
a) 3(b+1) = 24 (b)
b) c= ½ d +3 (d)
c) 9m + 4 =3m – 8 (m)
d) a = πr² (r)
7. The formula c= 5/9 (F ­ 32) is used to convert fahrenheit to centigrade.
In order to convert centigrade to fahrenheit it is easier to change the
subject of the formula to F.
Make F the subject and convert 100 degrees centigrade to fahrenheit.
8. Equations often have two unknowns. In the equation t = w + 10
· Find t when w = 5
· Find t when w = 35
· Find w when t = 5
9. In the equation: y = 3x – 5
· Find y when x = 3
· Find y when x = ­2
· Find x when y = 10
10.The air temperature T°C, outside an aircraft flying at a height of h feet
is given by the formula:
T = 26 ­ h
500
An aircraft is flying at a height of 27 000ft:
· What is the air temperature outside the aircraft?
· The temperature outside the aircraft is ­52°C. What is the height
of the aircraft?
Prepared with reference to and examples from:
Edexcel GCSE Mathmatics­ Int.(2001) Heinemann
Haylock,D(2001) Mathmatics Explained for Primary Teachers London: Paul Chapman Publishing
Suggate,J et al(2001) Mathmatical Knowledge for Primary Teachers London:David Fulton Publishers
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Formulae
For some equations you may need a common solution, these are called
simultaneous equations.
You can solve them by substitution.
Try and find values for x and y in the following example:
y = 2x + 1 and 3y – 4x = 13
Substitute 3 (2x + 1) – 4x =13
Remove brackets 6x + 3 – 4x = 13
Collect like terms 2x + 3 =13
Subtract 3 from both sides 2x = 10
Divide by 2 x = 5
Now find the solution for y by substituting in 5 for x in the first equation
y + 2x + 1:
Substitute x = 5 Y = 2x 5 + 1
Y = 10 +1
Y = 11
Check by substituting the values you have found into the equation containing
both unknowns, in this case x and y.
Correct!
3y – 4x = 3 x 11 – 4 x 5
= 33 – 20
= 13
11.Solve these pairs of equations:
a) y = 27 + 2x and y = 5x
b) y = 40 + 3x and y = 8x
c) y = 25 + 9x and y = 80 + 4x
Prepared with reference to and examples from:
Edexcel GCSE Mathmatics­ Int.(2001) Heinemann
Haylock,D(2001) Mathmatics Explained for Primary Teachers London: Paul Chapman Publishing
Suggate,J et al(2001) Mathmatical Knowledge for Primary Teachers London:David Fulton Publishers
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Formulae
Answers to questions
1) b = 2a + 3
2a + 3 = 131
2a = 131 –3
a = 128/2 a = 64
b – 3 = a
2
2) b = 3(a­1) ­ 7
3(a­1) – 7 = 8
3(a­1) = 15
a­1 = 5
a= 6
a = (b + 7) ¸ 3 + 1
3) 6m – 27 = 0
6m = 27
m = 27/6 m = 4.5
4) c 2 = 25
c = Ö25
c= 5 cm
5) M=miles K= Km
K = 1.609M
To get the number of Km
you multiply the number of
miles by 1.609.
6) (a) 3(b+1) = 24
3b + 3 = 24
b = 24 – 3
3
(b) c = ½ d + 3
½ d = c – 3
d = 2(C – 3)
(c) 9m + 4 = 3m – 8
9m = 3m ­ 12
6m = ­12
(d) a = Õr 2
r 2 = a
Õ
hhh
r = / a
Ö Õ
7) c = 5/9 (F­32)
5/9(F­32) = c
F – 32 = 9c
5
F = 9c + 32
5
F = (9 x 100) + 32
5
F = 900/5 + 32
F = 180 + 32
F = 212 degrees fahrenheit.
8) t = w + 10
when w = 5, t = 15
when w = 35, t = 45
when t = 5
5= w + 10
w = 5 – 10, w= ­5
9) y = 3x – 5
when x = 3, y = 4
when x= ­2, y = ­11
when y = 10
10 = 3x –5
3x = 15, x = 5
Prepared with reference to and examples from:
Edexcel GCSE Mathmatics­ Int.(2001) Heinemann
Haylock,D(2001) Mathmatics Explained for Primary Teachers London: Paul Chapman Publishing
Suggate,J et al(2001) Mathmatical Knowledge for Primary Teachers London:David Fulton Publishers
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Formulae
10) T = 26 – (h/500)
T = 26 – (27,000/500)
T = 26 – 54
T = ­28 degrees centigrade
T = 26 – (h/500)
T + (h/500) = 26
h/500 = 26 – T
h = 500(26 – T)
h = 13000 ­ 500T
h = 13000 ­ (500 x (­52))
h = 13000 + 26000
h = 39,000 feet
11)
a) y = 27 + 2x and y = 5x
5x = 27 + 2x
3x = 27
x = 9
y = 5x
y = 5 x 9
y = 45
b) y = 40 + 3x and y = 8x
8x = 40 + 3x
5x = 40
x = 8
y = 8x
y = 8 X 8
y = 64
c) y = 25 + 9x and y = 80 + 4x
80 + 4x = 25 + 9x
5x = 55
x = 11
y = 25 + 9x
y = 25 + (9 x 11)
y = 124
Prepared with reference to and examples from:
Edexcel GCSE Mathmatics­ Int.(2001) Heinemann
Haylock,D(2001) Mathmatics Explained for Primary Teachers London: Paul Chapman Publishing
Suggate,J et al(2001) Mathmatical Knowledge for Primary Teachers London:David Fulton Publishers
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