Example 1: Solve the system of linear equations.
2x + 3y = 8
3x  y = 5
Solution to example 1

multiply all terms in the second equation by 3
2x + 3y = 8
9x  3y = 15

add the two equations
7x = 7

Note: y has been eliminated, hence the name: method of elimination

solve the above equation for x
x = 1

substitute x by 1 in the first equation
2(1) + 3y = 8

solve the above equation for y
2 + 3y = 8
3y = 6
y = 2

write the solution to the system as an ordered pair
(1,2)

check the solution obtained
first equation: Left Side: 2(1) + 3(2)= 2 + 6 = 8
Right Side: 8
second equation: Left Side: 3(1)(2)=32=5
Right Side: 5

conclusion: The given system of equations
is consistent and has the ordered pair, shown below,
as a solution.
(1,2)
Matched Exercise 1: Solve the system of linear equations.
x + 3y = 11
4x  y = 11
Example 2: Solve the system of linear equations.
4x  y = 8
8x + 2y = 5
Solution to example 2

multiply all terms in the first equation by 2
8x  2y = 16
8x + 2y = 5

add the two equations
0x + 0y = 11 or 0 = 11

Conclusion: Because there are no values of x
and y for which 0x + 0y = 11, the given system of equations has
no solutions. This system is inconsistent.
Matched Exercise 2: Solve the system of linear equations.
2x + y = 8
6x  3y = 10
Example 3: Solve the system of linear equations.
2x  3y = 8
4x + 6y = 16
Solution to example 3

multiply all terms in the first equation by 2
4x  6y = 16
4x + 6y = 16

add the two equations
0x +0y = 0 or 0 = 0

Conclusion: The system has an infinite
number of solutions. The solution set consists of all ordered
pairs satisfying the equation 2x  3y = 8. This system is consistent.
Matched Exercise 3: Solve the system of linear equations.
x  2y = 3
3x + 6y = 9
References and links related to systems of equations.