Download Solving equations - LEARN MATHS by Mr. Dalaba M.Sc B Ed

L M
earn
aths
MATHEMATICS
For
GCSE
Solving Linear Equations
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Denis Dalaba MSc, B.Ed.(Hons)
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by Denis Dalaba
Page 1
Solving linear equations
Some examples of linear equations are
An equation is a mathematical statement stating that two expressions are equal.
The unknown number is a particular number that can be worked out.
To solve an equation means finding a …………………………. that makes the
equation a true statement.
This can be done by mental inspection or using algebraic manipulation.
Solving equations by mental inspection or trial and improvement
Very simple equations can often be solved mentally or by trial and improvement
Example
Solve the equations
i.
x+3=5
ii. 2n = 20
iii.
v
8
4
Solutions
Stop the video and Try Exercise 1
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Page 2
Algebraic method
We are now going to learn a systematic procedure (algebraic way) of
solving equations. The algebraic method can be used to solve both simple
and complex equations.
Introduction
Solving equations depend on two maths properties:
a. the ………………………………………..
b. the ………………………………………..
a. ………………………………………………………
=
In the figure above, the left hand side (LHS) and right hand side (RHS) both
have 10 rectangles each.
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Page 3
i.
Subtracting ……from both sides will result in both being left with...
ii.
Adding ………….. to both sides will result in both having ……...
iii.
Multiplying both sides by ……… will result in both having …………
iv.
Dividing both sides by ……… will result in both being left with …….
In algebra whenever we perform an operation on the left hand side
(LHS) of an equal sign we must do the same to the right hand side
(RHS) to maintain the equivalence (that is to ‘balance’ the equation).
b. Inverse operations (Reverse operation) or ‘undo’ property
Or
Additive Inverse
1. The additive inverse of ‘+ 5’ is ……..
2. The additive inverse of ‘ - 9’is ……….
1. To undo ‘add 5’ we ………...
2. To undo ‘subtract 9’we ………
Multiplicative Inverse (reciprocal)
3. The multiplicative inverse of ‘×7 ’is ……..
3. To undo ‘multiply by 7’we………
4. The multiplicative inverse of ‘÷ 10’is ……...
4. 4. To undo ‘divide by 10’ we
Note
In algebra, x  5 is written as
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x
5
by Denis Dalaba
Page 4
Fact 1
i.
3+(
)=
ii.
-2+
=
iii. a + (
)=
Fact 2
By the property of inverse (or reciprocals) we know that
a.
5

=
b.
1

7
=
2
c. ×
9
=
Solving equations
We solve equations by reducing (restating) the equation to an equivalent
equation of the form x = a where a is an integer.
That is x = some number.
Example
2x + 5 = 17
We can work out mentally that
x=6
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Page 5
Solving equations involving additive inverse
We are now going to apply additive inverse with the balancing
property to solve equations
Example
Solve the equations
i.
ii. y – 9 = 21
x+3=7
Solution
Stop the video and Try Exercise 2
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Page 6
Solving equations using multiplicative inverse
We are now going to apply multiplicative inverse (also called the reciprocal)
with the balancing property to solve equations.
Type 1- Multiplication
Solve the equations
i.
15m = 45
ii. 12b = 6
iii. 10 = 5t
Strategy: To solve an equation in which the variable is multiplied by a constant,
we start by multiplying both the LHS and the RHS by the reciprocal of the
constant.
Stop the video and Try Exercise 3
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Page 7
Type 2 – Division
Examples
Solve the equations
i.
𝑥
4
= 20
1
ii. c = 2
9
iii.
1
0.5
𝑦 = 16
Stop the video and Try Exercise 4
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by Denis Dalaba
Page 8
Type 3- Multiplication & Division
Example
Solve
i.
2
3
y = 12
Stop the video and Try Exercise 5
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Page 9
Solving equations using combination of strategies
We are now going to use a combination of strategies (add, subtract, multiply
and or divide) to solve equations.
Examples
Solve the following
a. 5 x + 3 = 18
b.
4w – 7 = 21
c.
2
7
y + 1 = 11
Solution
Stop the video and Try Exercise 6
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by Denis Dalaba
Page 10
Solving equations involving brackets (parentheses)
Example
Solve the following
i.
2(x + 5) = 21
ii.
4(y +2) = 20
iii.
7 (x - 1 ) = 12
iv.
5(2n + 7 ) + 3 = 78
Strategy:

Expand to get rid of the parentheses(brackets)

Group and simplify like terms

Do the relevant inverse operations
Note:
In some cases expanding the bracket first in not necessary
500 (2x - 1) = 1000
Start by dividing by the HCF
Stop the video and Try Exercise 7
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by Denis Dalaba
Page 11
Solving equations involving fractions
Examples
Solve the following
i.
2
x4
5
ii.
iii.
ii.
3t 9

11 4
3
x5
7
iv.
7 6m

3
5
Stop the video and Try Exercise 8
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Page 12
Solving equations with the variable (unknown) on both the LHS and RHS
Examples
Solve the equations
i.
7x + 1 = x + 13
ii.
5 + 3y = 2y -7
ii.
5b + 1 = 16 - 2b
Stop the video and Try Exercise 9
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Page 13
Solving equations containing exponentials and radicals
Examples
i.
iii.
x2  9
ii.
2y 2 = 32
3x  5
Stop the video and Try Exercise 10
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Page 14
Mixed questions
Find x?
260
2x
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x
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Page 15