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.

• Gaussian Elimination Solver Calculator for a 3 by 3 Systems of Equations.
  
• Solver Calculator to Solve 3 by 3 Systems of Equations Using Cramers' Rules.
  
• Systems of Equations Calculator and Solver.
  
• Systems of Linear Equations - Graphical Approach
  
• Solve Equations, Systems of Equations and Inequalities.