# 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.