1. Systems of Linear Equation
In real life, different persons have different personalities, characteristics, likes,
dislikes, and capabilities. However, there are times when two persons have common likes
and dislikes in which case we say hat they are consistent. There are also times when we
find two persons whose vibrations are dependent while two persons who do not have
anythingin common at all are inconsistent.
A "system" of equations is a set or collection of equations that you deal with all
together at once. Linear equations (ones that graph as straight lines) are simpler than non-linear
equations, and the simplest linear system is one with two equations and two
variables.
2. Problem Solving About System of Linear Equation
Number Problems:
Example 1.
Two numbers are in the ratio 4:3. Find the numbers if their sum is 84.
Solution:
To find the required numbers, we proceed as follows:
Let x = smaller number
84 x= the greater number
The two numbers are in the ratio of 4:3, so
4
3
=
84 − 푥
푥
4푥 = 3(84 − 푥)
4푥 = 252−3푥
4푥 + 3푥 = 252
7x = 252
7푥
252
=
7
7
x = 36
84−푥 = 48
Therefore, the numbers are 36 and 48.
Cheking, we have
36+48 = 84 and
48
36
=
4
3
84 ≛ 84
Example 2.
Find the two consecutive odd integers whose sum is 44.
Solution:
Let x = the first odd integer
x + 2 = the second integer
Equation: x+(x+2) = 44
Solving the Equation, we have
x+(x+2) = 44
2x+2 = 44
x = 21 (the first odd integer)
x+2 = 23 (the second odd integer)
Therefore,the consecutive odd interger are 21 and 23.
Cheking:
21+23 = 44
44 ≛ 44
3. Example 3.
The difference of two numbers is 14. Twice the smaller number is 5 less than the
larger number. Find the numbers.
Solution:
Let x = the larger number
Let y = the smaller number
The difference of the two numbers is 14. So,
푥 − 푦 = 14
Twice the smaller number is 5 less than the larger number. Thus,
2푦 = 푥 − 5
We can now consider the system of equations
푥 − 푦 = 14 → (1)
2푦 = 푥 − 5 → (2)
Rewriting the equations, we have
푥 − 푦 = 14 → (1)
푥 − 2푦 = 5 → (2)
Subtracting (2) from (1), we obtain
푦 = 9
Substituting y = 9 to (1), we get
푥 − 9 = 14
푥 = 23
Therefore, the numbers are 23 and 9.
Checking:
23 − 9 = 14
14 ≛ 14
23 − 2(9) = 4
4 ≛ 4
Example 4.
The numerator of a fraction is 3 less than the denominator. If the numerator and
denominator are each increased by 1 the value of the fraction becomes
3
4
. What is the
original fraction?
Solution:
We can represent the unknown values as follows:
Let x = the denominator
푥 − 3 = 푡ℎ푒 numerator
Based on the condition of the problem, the equation is
(푥−3)+1
푥+1
= 3
4
Simplify the equation and solving, we obtain
푥−2
푥+1
=
3
4
4푥 − 8 = 3푥 + 3
4푥 − 3푥 = 3 + 8
