Example:1
Solve the following linear equations in three variables:
2x + y z = 4 (1)
x + y + z = 10 (2)
2z = 4 (3)
Solution:
From equation 3 we have 2z = 4 then z = 4/2 = 2.
Therefore z = 2.
substitute z = 2 in equation 1 and equation 2.
i.e 2x + y z = 4 => 2x + y  2 = 4 => 2x + y = 6.
and x + y + z = 10 => x + y + 2 = 10 =>x + y = 8
Solve these equations
2x + y = 6 and x + y = 8
2x + y = 6
x  y = 8

x = 2
i.e x = 2.
Substituting values x = 2 and z = 2 in equation 2.
i.e x + y + z = 10
=> 2 + y + 2 = 10
i.e y = 10
Therefore x = 2, y = 10 and z = 2
Verification:
If we substitute x = 2, y = 10 and z = 2 in 2x + y  z = 4, then we have
2x + y  z
= 2(2) + 10  2
= 4 + 10  2
= 6 + 10
= 4
Hence verified.
Example:
Solve the following linear equations in three variables.
a + b = 12 (1)
b + c = 6(2)
c + a = 8(3)
Solution:
From equation 1 we have,
a + b = 12 => b = 12  a
substitute b = 12  a in equation 2
12  a + c = 6 =>  a + c =  6 =>  a + c =  6.
Also equation 3 is c + a = 8.
Add both equations.
 a + c =  6 (2)
a + c = 8(3)

2c = 2
therefore c = 1
substituting c = 1 in equation a + c = 8 we get
a + c = 8 => a + 1 = 8 => a = 7.
substituting c = 1 and a = 7 in equation 1(2 or 3) we get
a + b = 12 => 7 + b = 12 => b = 5.
Therefore a = 7, b = 5 and c = 1.
Verification:
If we substitute a = 7, b = 5 and c = 1 in a + b = 12 and b + c = 6, then we have
a + b = 7 + 5 = 12
and b + c = 5 + 1 = 6.
Hence verified.
Example:
Solve the following linear equations in three variables:
a + b + c = 10 (1)
a + 2b + 3c = 5 (2)
2a + b + 5c = 4(3)
Solution:
a + b + c = 10equation 1
a = 10  b  c
a + 2b + 3c = 5 equation 2
(10  b  c) + 2b + 3c = 5 => 10  b  c + 2b + 3c = 5 => 10 + b + 2c = 5 => b + 2c = 5
2a + b + 5c = 4equation 3
2(10  b  c) + b + 5c = 4 =>20  2b  2c + b + 5c = 4 => 20  b + 3c = 4 => b + 3c = 16
b + 2c = 5
b + 3c = 16 solving these equations we will find b and c.
b + 2c = 5
b + 3c = 16

0 + 5c = 21
Hence c = 21/5
substituting in 1 we get a + b = 1/5
Directions: Solve the following equations with three variables. Also write at least 5 examples of your own.
