Solve Systems of Equations - Tutorial

This is a tutorial on solving 2 by 2 systems of linear equations. Detailed solutions and explanations are provided.



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.


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Updated: 2 April 2013

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