The document provides a series of examples and explanations on how to solve equations with one variable, illustrating the principle that maintaining equality in an equation requires performing the same operation on both sides. Several example equations are presented, including ways to isolate the variable and confirm the solution. It also touches on the effects of transferring constants across the equals sign and simplifying expressions.

1. Solutions to
Equations
Grade 8 - Mathematics

2. Expression
Equation
2x + 5
2x + 5 = 9
there are no "sides"
and no = sign
has a left side, a right side
and an equals sign
separating the sides

3. 2x + 5 = 9
has a left side, a right side
and an equals sign
separating the sides
Equation
Left side Right
side
Equals
sign

4. How do you solve an equation with
one variable?
For example, if
you have an equation
3 – 5x = 18
how do you know
how to solve for x?

5. Solution to one equation
For example,
In the equation
x + 2 = 3
the solution is x =
since 1 + 2 = 3
- it is the values that make the equation true
1

6. x – 3 = 5
What is the solution to this equation?
If the same number
was added to both
sides of an equation,
the sides remain
equal
A. P. E.

7. x – 3 = 5
What is the solution to this equation?
If a negative integer
will be transferred to
the other side, it will
become positive.
x = 5 + 3
x = 8
A. P. E.

8. x + 3 = 5
What is the solution to this equation?
when both sides of an
equation have the same
number subtracted from
them, the sides remain
equal
S. P. E.

9. x + 3 = 5
What is the solution to this equation?
If a positive integer
will be transferred to
the other side, it will
become negative.
x = 5 – 3
x = 2
S. P. E.

10. What is the solution to this equation?
both sides of an
equation are
multiplied by the
same number.
a = 30
M. P. E.a
2
= 15
a
2
= 152 2

11. 3x = 9
What is the solution to this equation?
To eliminate the
factors, Multiply it by
the number itself.x = 3
3 3
D. P. E.

12. What is the solution to this equation?
3 – 5x = 18
Left side Right
side
Equals
sign
Let us retain the
solution to equation
on the left side.

13. What is the solution to this equation?
Transfer the constant
to the right side
3 – 5x = 18

14. – 5x = 18
What is the solution to this equation?
Transfer the constant
to the right side
3 –

15. – 5x = 18
What is the solution to this equation?
Simplify the right side
3–
– 5x = 15

16. – 5x = 18
What is the solution to this equation?
Remove the
coefficient of x by
3–
– 5x = 15
dividing –5 to both
sides
– 5 – 5

17. – 5x = 18
What is the solution to this equation?
Simplify
both sides
3–
– 5x = 15
– 5 – 5
x = – 3

18. What is the solution to this equation?
Replace x
with -3
If x = –3
3 – 5x = 18
3 – 5(–3) = 18
3 – (–15) = 18
3 + (+15) = 18
18 = 18

19. What is the solution to this equation?
4x + 8 = 16 – 5x – 1 + 4x = 3 2(x – 4) = 3x – 9

20. What is the solution to this equation?
x – x 1 x – 3 1 x+4=
3 4 6
=
3 2 6
3x –+

21. What are the only two countries in the entire world
that do not sell Coca-Cola?
11 4 10 2 5 6 4 10 8 1 1 11 12 3 9 7 1
1. Solve for x ∶ 𝟑𝒙 + 𝟓 + 𝟔𝒙 − 𝟕 = 𝟐𝟓

22. What are the only two countries in the entire world
that do not sell Coca-Cola?
11 4 10 2 5 6 4 10 8 1 1 11 12 3 9 7 1
2. Solve for w ∶ 𝟏𝟑𝒘 − 𝟐 𝟒𝒘 + 𝟏 = 𝒘 − 𝟓𝟖

23. What are the only two countries in the entire world
that do not sell Coca-Cola?
11 4 10 2 5 6 4 10 8 1 1 11 12 3 9 7 1
12. Solve for x: 𝐱 − 𝟓/ 𝟐 = 𝟗