solving equations

1. D. J. Jayamanne
NSBM 1
Solving Equations

2. Equations
An equation links an algebraic expression and a number, or two algebraic
expressions with an equals sign.
For example: 𝑥 + 5 = 12 is an equation.
In an equation the unknown usually has a particular value. Finding the
value of the unknown is called solving the equation.
𝑥 + 5 = 12
𝑥 = 12 − 5
𝑥 = 7
NSBM 2

3. • Always check the solution to an equation by substituting the solution
back into the original equation.
e.g. 𝑥 + 5 = 12 ; 𝑥 = 7 is a solution.
• If we substitute x = 7 back into x + 5 = 12 we have
7 + 5 = 12
NSBM 3

4. Examples
Example-1
5𝑥 = 45
𝑥 =
45
5
𝑥 = 9
Example-2
17 = 6 + 𝑥
17 − 6 = 𝑥
11 = 𝑥
𝑥 = 11
NSBM 4
Check: 9 × 5 = 45
We always write the letter before the equals sign.
Check: 6 + 11 = 17

5. Examples
Example-3
𝑥
5
= 3
𝑥 = 3 × 5
𝑥 = 15
Example-4
3𝑥 − 4 = 14
3𝑥 = 14 + 4
3𝑥 = 18
𝑥 =
18
3
𝑥 = 6
NSBM 5
Check:
15
5
= 3
Check: 3 × 6 − 4 = 14

6. Solving equations by transforming both sides
Solve this equation by transforming both sides in the same way:
NSBM 6
Add 1 to both sides.
Multiply both sides by 4.
m = 12
– 1 = 2
m
4
= 3
m
4
+1 +1
×4 ×4

7. Solving with brackets
• Equations can contain brackets. For example:
NSBM 7
2(3x – 5) = 4x
To solve this we can
Multiply out the brackets: 6x –10 = 4x
+10 +10
Add 10 to both sides: 6x = 4x + 10
−4x −4x
Subtract 4x from both sides: 2x = 10
÷ 2 ÷ 2
Divide both sides by 2: x = 5

8. Questions
Q1. Solve the following equations:
NSBM 8
i. 3𝑎 = 18
ii. 8 = 2𝑎
iii. 2𝑎 + 1 = 7
iv. 3𝑎 + 2 = 17
v. 2𝑎 = 60
vi. 6 = 2𝑎
vii. 3𝑎 + 2 = 14
viii. 4𝑎 + 6 = 50
ix. 5𝑎 = 40
x. 2𝑎 = 1
xi. 4𝑎 + 7 = 19
xii. 6𝑎 + 5 = 41

9. Questions
Q2. Solve the following equations:
NSBM 9
i. 5𝑥 − 3 =7
ii. 3𝑥 − 4 = 11
iii. 7𝑥 + 3 = 24
iv. 6𝑥 + 5 = 6
v. 9𝑥 + 1 = 100
vi. 3𝑥 − 5 = 10
vii. 3 + 2𝑥 = 15
viii. 5 + 3𝑥 = 11
ix. 8 + 4𝑥 = 8
x. 14 = 3𝑥 − 1
xi. 31 = 7𝑥 + 3
xii. 100 = 5𝑥 − 5

10. Questions
Q3. Solve the following equations:
NSBM 10
i. 2(𝑥 + 1) = 10
ii. 2(𝑥 + 3) = 12
iii. 3(𝑥 + 4) = 21
iv. 3(𝑥 − 2) = 12
v. 3 2𝑥 + 1 = 9
vi. 4 𝑥 − 2 = 8
vii. 5 𝑥 + 1 = 5
viii. 2 3𝑥 − 1 = 10
ix. 2 3𝑥 + 2 = 10
x. 2(𝑥 + 3) = 7
xi. 4 𝑥 + 1 = 5
xii. 6(𝑥 + 2) = 13

11. Questions
Q4. Solve the following equations:
NSBM 11
i. 5(𝑎 + 1) = 20
ii. 3(𝑡 − 1) = 18
iii. 4(𝑏 + 3) = 20
iv. 3(2y + 3) = 10
v. 14 = 2(3a + 1)
vi. 16 = 4(𝑛 − 2)
vii. 18 = 2(2m + 3)
viii. 5(2𝑥 + 2) = 10
ix. 3(2n − 7) = 3
x. 8(2 + 𝑥) = 24
xi. 10 3 + 𝑥 = 100
xii. 5 1 + 2𝑥 = 20

12. Questions
Q5. Solve each equation:
NSBM 12
i.
𝑥−7
3
= −12
ii.
𝑦−1
4
= −2
iii. 6 =
𝑚
7
− 3
iv.
2
𝑥+1
=
5
2
v. 2 +
𝑦
5
= −8
vi. −14 = −5 + 3c
vii. 10 =
𝑥
4
− 8
viii.
𝑎+10
2
= 4
ix. 7
1
2
=
𝑥+3
2
x.
𝑔−3
3
=
5
3

13. Questions
Q6. Solve each equation:
NSBM 13
i.
2+𝑦
3
= −1
ii. −24 = −10𝑡 + 3
iii. 10 = 0.3𝑥 − 9.1
iv.
1
2
=
1
2
𝑐 − 2
v.
𝑥−3
3
= −4
1
2
vi. 9.4 = −d + 5.6
vii.
𝑑+17
2
= 5
1
3
viii. 2.4 + 10m = 6.89
ix.
1
5
t − 3 = −17

14. Constructing an equation
1. A delivery person uses a service elevator to bring boxes of books up to
an office. The delivery person weighs 160 lb and each box of books
weighs 50 lb. The maximum capacity of the elevator is 1000 lb. How many
books of boxes can the delivery person bring up at one time?
2. You have $ 16 and a coupon for a $5 discount at a local super market. A
bottle of olive oil costs $7. How many bottles of olive oil can you buy?
3. Dominique paints faces at an annual carnival. Her goal this year is to
earn a profit of $100. She spends $15 on supplies and will work for 2.5h.
How much will she need to earn in dollars per hour in order to reach her
goal?
NSBM 14

15. Constructing an equation
4. Two college friends rent an apartment. They have to pay the landlord
two months’ rent and a $500 security deposit when they sign the lease.
The total amount they pay the landlord is $2800. What is the rent for one
month?
5. Ben started with 3 packets of sweets and ate 11 sweets. Lucy started
with 2 packets of sweets and ate 3 sweets. Assuming Ben and Lucy have
the same number of sweets find the number of sweets in a packet?
NSBM 15

16. Constructing an equation
6. Geometry: The perimeters of the triangles shown are equal. Find the
side lengths of each triangle.
NSBM 16
2x+3
2x+4
2x
2x 2x
3x+1

17. Constructing an equation
7. Health Clubs: One health club charges a $50 sign-up fee and $65 per
month. Another club charges a $90 sign up-fee and $45 per month. For
what number of months is the cost of the clubs equal?
8. Business: A small juice company spends $1200 per day on business
expenses plus $1.10 per bottle of juice they make. They charge $2.50 for
each bottle of juice they produce.
a. How many bottles of juice must the company sell in one day in order to
cover its daily costs?
b. How much bottles must the company sell to achieve a profit of $200
daily?
NSBM 17

18. Constructing an equation
9. Travel: Suppose a family drives at an average rate of 60 mi/h on the
way to visit relatives and then at an average rate of 40 mi/h on the way
back. The return trip takes 1 h longer than the trip there.
a. Let d be the distance in miles the family travelled to visit their
relatives. How many hours did it take to drive there?
b. In terms of d, how many hours did it take to make the return trip?
c. Write and solve an equation to determine the distance the family
drove to see their relatives.
d. What was the average rate for the entire trip?
NSBM 18

19. Constructing an equation
10. Printing: Pristine Printing will print business cards for $0.1 each plus a
set up charge of $15 . The Printing Place offers business cards for $.15
each with a set up charge of $10. What number of business cards costs the
same from either printer.
NSBM 19

20. Identities and Equations with no solutions
How can you tell how many solutions an equation has?
NSBM 20
If you eliminate the variable in the process of solving,
the equation is either an identity with infinitely many
solutions or an equation with no solution.

21. Identities and Equations with no solutions
What is the solution of each equation?
Example-1 :
10𝑥 + 12 = 2 5𝑥 + 6
10𝑥 + 12 = 10𝑥 + 12
Because 10𝑥 + 12 = 10𝑥 + 12 is always true, there are infinitely many
solutions of the equation. The original equation is an identity.
NSBM 21

22. Identities and Equations with no solutions
What is the solution of each equation?
Example-2 :
𝟗𝒎 − 𝟒 = −𝟑𝒎 + 𝟓 + 𝟏𝟐𝒎
9𝑚 − 4 = −3𝑚 + 5 + 12𝑚
9𝑚 − 4 = 9𝑚 + 5
−4 = 5 ✖
Because −4 ≠ 5, the original equation has no solution.
NSBM 22

23. Q
Match each equation with the appropriate number of solutions
1. 3𝑦 − 5 = 𝑦 + 2𝑦 − 9 A. Infinitely many
2. 2𝑦 + 4 = 2(𝑦 + 2) B. One solution
3. 2𝑦 − 4 = 3𝑦 − 5 C. No solution
NSBM 23

24. Questions
Q6. Solve each equation:
NSBM 24
i.
2+𝑦
3
= −1
ii. −24 = −10𝑡 + 3
iii. 10 = 0.3𝑥 − 9.1
iv.
1
2
=
1
2
𝑐 − 2
v.
𝑥−3
3
= −4
1
2
vi. 9.4 = −d + 5.6
vii.
𝑑+17
2
= 5
1
3
viii. 2.4 + 10m = 6.89
ix.
1
5
t − 3 = −17