

Linear
equations 
Linear
equation 
Solving linear equations 
Applications
of linear equations, word problems 
Numbers word problem 
Geometry and physics word problems






Linear
equation 
An equation is linear if it can be written in the form 
ax =
b,
where
x
is the variable and a and
b are constants. 
A linear, means the
variable x appears only to the first power. 
The solution of an
equation is the value of the variable x that makes the equation a
true statement. 

Solving linear equations 
We solve an equation by transforming it through a
sequence of equivalent equations, statement by statement, until x
finally is isolated on the left. 
Two equations that have the same
solutions are called equivalent. 
Shift a number from one side of an
equation to the other, by writing it on the other side with the inverse
operation. 
When the operations are addition or subtraction a term
may be shifted to the other side of an equation by changing its sign. 
After solving an equation check the solution by substituting them in
the original equation. 

Examples:
a)
1

(2

x)
= 2

(3x
+
1) 
1

2
+
x
= 2

3x

1 
x
+
3x
= 2 +
2 
4x
= 4

¸
4 
x
= 1 




Applications of the linear
equations,
word
problems

Word problems  translate the words into mathematics 
Numbers word
problem 
Example:
The sum of two digits of a twodigit number is 10. If the digits are
reversed, the new number is 
four less then five times the original number. Find twodigit number. 
Solution:
Let
x be
tens place value digit then,
10

x is units digit, thus given number is
10x
+ (10

x). 
By reversing digits we get
10
· (10

x)
+ x. 
Hence
given problem yields: 10
· (10

x)
+ x
= 5
· [10x
+ (10

x)]

4 
100

10x
+ x
= 5
· [9x
+ 10]

4 
100

9x
= 45x
+ 50

4 
54x
= 54
 ¸
54 
x
= 1
thus,
the twodigit number is:
10x
+ (10

x)
=19. 

Geometry and physics word problems

Example:
A rectangle has the perimeter
90 cm and the width 13 cm less then the length. 
What is the area of the rectangle? 


a
= b
+ 13
= 16
+
13
= 29
cm 
A
= a
· b
= 29
·
16
= 464
cm^{2} 

the perimeter
P
of a rectangle: 
P
= 2
·
(a
+ b)
= 90 
2
·
(b + 13
+ b)
= 90
 ¸
2 
2b
+ 13
= 45 
2b
= 32 
b
= 16
cm 



Example:
After a car A started traveling at a speed of 90 km/h, halfhour later
from the same place in the 
same direction starts car B traveling at 120 km/h. When will car B reach the car A? 

Solution:
The cars are traveling the same distance
d
= r
· t .
distance = rate x time 
The car A takes x hours, and the car B,
x
 1/2
hours. 
Thus, 90
· x =
120
· (x
 1/2)
 ¸
30 
3x
=
4x
 2 
x
=
2
hours, the car B will reach the car A after
x
 1/2 =
2
 1/2 =
3/2 hours. 









Beginning
Algebra Contents C 



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