__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)=-3-2=-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.