Solving Linear Equations

1. SOLVINGLINEAR
EQUATIONS
PREPARED BY:
ALONA HALL

2.  A layman’s definition of the
term ‘equation’will be
provided
 The ‘balancing method’ for
solving an equation will be
explained
 A shorter method for
solving an equation will be
proposed
 The method for checking
solution of the equation
will be presented
The
objectives of
this lesson
are as
follows:

3. The word ‘solve’ in
layman’s terms means
‘to find the unknown/s
in an equation’
NB:
only equations can be solved So what are
equations?

4. An equation is a mathematical
statement in which two things are
equal. It consists of two expressions,
one on each side of an 'equal' sign
It is like a balance scale…both sides must value
the same thing in order for it to be balanced.
Additionally, an equations is a mathematical
sentence that is given with an equal sign.

5. Like a balance scale, an equation has two sides, the Left Hand Side (LHS) and the Right
Hand Side(RHS).
The equal sign separates both sides
LHS RHS
𝟑𝒒 − 𝟒 = −𝟏𝟑
𝟐𝒙 − 𝟖 = 𝟐𝟎 (x + 2)
𝒙
𝟐
−
𝟏
𝟒
= 𝟏𝟑

6. Thebalancescalehere
representsacasein
whichtwopoundsof
flourisonbothsides.
Asaresult,itbalances
6

7. If,however,onepound
offlourwasremoved
fromtherighthand
sidethenthescale
wouldlooklikethis:
7

8. Ifthescaleisto
balanceagain,
then
8
We would need to
remove one pound of
flour from the left side
return the pound
on the right sideOR

9. OR
9
One pound is also
removed from the
left side
One pound is
returned to the
right side

10. Theunderlyingprinciple
isthis:
WHATEVERISDONETO
ONESIDEMUSTBE
DONEONTHEOTHER
SIDEINORDERTO
BALANCE
10

11. Let’slookatan
example
3x + 8 = 23
11
Firstly, we must consider
what happened to the
unknown, x

12. 12
Reversetheprocess
(Undowhatwasdone
inreverseorder)

13. WEWILLNOWLOOKAT
THEBALANCING
METHOD
13

14. 14
𝟑𝒙 + 𝟖 = 𝟐𝟑
‘
𝒂𝒅𝒅𝒊𝒏𝒈 𝟖′
𝒘𝒂𝒔 𝒕𝒉𝒆 𝒍𝒂𝒔𝒕 𝒕𝒉𝒊𝒏𝒈 𝒅𝒐𝒏𝒆
𝒓𝒆𝒗𝒆𝒓𝒔𝒊𝒏𝒈 𝒕𝒉𝒆 𝒍𝒂𝒔𝒕 𝒕𝒉𝒊𝒏𝒈 𝒎𝒆𝒂𝒏𝒔 𝒔′
𝒖𝒃𝒕𝒂𝒄𝒕𝒊𝒏𝒈 𝟖
𝒃𝒖𝒕……
𝒘𝒉𝒂𝒕𝒆𝒗𝒆𝒓 𝒊𝒔 𝒅𝒐𝒏𝒆 𝒐𝒏 𝒐𝒏𝒆 𝒔𝒊𝒅𝒆 𝒎𝒖𝒔𝒕 𝒃𝒆 𝒅𝒐𝒏𝒆 𝒕𝒐 𝒕𝒉𝒆 𝒐𝒕𝒉𝒆𝒓 𝒔𝒊𝒅𝒆 𝒊𝒏 𝒐𝒓𝒅𝒆𝒓 𝒕𝒐 𝒃𝒂𝒍𝒂𝒏𝒄𝒆
𝒔𝒐 𝒘𝒆 𝒈𝒆𝒕
𝟑𝒙 + 𝟖 − 𝟖 = 𝟐𝟑 − 𝟖

15. Now we have
3x = 15.
The first thing that was done to x was ‘multiply by 3’.
Thus, reversing the first thing means ‘divide by 3’
but….
whatever is done on one side must be done on the other
side in order to balance
so we get
15
𝟑𝒙
𝟑
=
𝟏𝟓
𝟑

16. The result will be
x = 5
known as the solution.
At this point we know the number that was
unknown
16

17. Anotherexampleusingthe
balancingmethod:
17

18. 𝒙
𝟒
- 5 = -3
𝒙
4
- 5 + 5 = -3 + 5
𝒙
4
= 2
4 x
𝐱
4
= 4 x 2
x = 8
18

19. 𝑳𝒆𝒂𝒓𝒏 𝒕𝒉𝒆 𝒔𝒉𝒐𝒓𝒕 𝒎𝒆𝒕𝒉𝒐𝒅.
Here’s the example:
Solve 5t – 8 = 17
Firstly, you must observe all the
operations done on the unknown, t.
Then undo them in reverse order
19

20. 5t – 8 = 27
The last thing was ‘subtract 8’
so ‘add 8’but show it on the other side
5t = 27 +8
5t = 35
The reverse of ‘multiply by 5’is ‘divide by 5’.
Show it on the other side
x =
𝟑𝟓
𝟓
x = 7 is the solution 20

21. Another example and
solution
21

22. 𝟑𝒑 + 𝟗 = −𝟒
3p = -4 - 9
3p = -13
p =
−𝟏𝟑
𝟑
22

23. Solving the Equation
and Checking the
Solution
23
Find the solution of the
following equation”
𝑚
3
- 1 = -5

24. 𝒙
3
= -5 + 1
𝒙
𝟑
= - 4
x = -4 x 3
x = - 12
The solution is x = -12
24

25. 𝑪𝒉𝒆𝒄𝒌:
−𝟏𝟐
𝟑
− 𝟏 = - 5
-4 – 1 = - 5
-5 = -5
LHS = RHS
Both sides balance. Thus, the solution is
CORRECT!
25

26. Suggestion:
Reviewthislessoninordertoremindyourselfofthe
stepsinvolvedinsolvinganequation,especiallythe
shortmethod.Trysomeonyourown.
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