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.