Download 4.1 Systems of Linear Equations in Two Variables Worksheet

4.1 Systems of Linear Equations
Solving a Linear System by the Elimination Method
1. Write both equations in standard form Ax + By = C.
2. Make the coefficients of one pair of variable terms opposites. Multiply one or both equations by the
appropriate numbers so that the sum of the coefficients of either the x or y terms is 0.
3. Add the new equations to eliminate a variable. The sum should be an equation with just one variable.
4. Solve the equation from Step 3 for the remaining variable.
5. Find the other variable. Substituted the result of Step 4 into either of the original equation and solve for the
other variable.
6. Check the ordered pair solution in both of the original equations.
Solve the system.
Example 1
Example 2
Solve each system.
x + y = –5
–2x + y = 1
1.
2. 4x – 5y = –11
x + 2y = 7
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
x-value: _______
y-value: _______
Solution: ( , )
3. 5x – 4y = 9
x – 2y = –3
4. x – 4y = – 4
3x + y = 1
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
x-value: _______
y-value: _______
Solution: ( , )
5.
8x + 4y = 0
–4x + 2y = –2
6.
x+y=4
–2x + y = –2
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
x-value: _______
y-value: _______
Solution: ( , )
7. –3x + y = –5
x + 2y = 0
8. 3x + y = –7
x – y = –5
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
x-value: _______
y-value: _______
Solution: ( , )
9. 3x – 2y = 0
–9x – 8y = –7
10. 3x – y = 10
2x + 5y = 1
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
Example 3 Solve the system of equations.
x-value: _______
y-value: _______
Solution: ( , )
11. 3x + 4y = – 6
5x + 3y = 1
12. 4x + 3y = 1
3x + 2y = 2
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
x-value: _______
y-value: _______
Solution: ( , )
13. 3x + 3y = 0
4x + 2y = 3
14. 3x – 5y = 7
2x + 3y = 30
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
x-value: _______
y-value: _______
Solution: ( , )
15. 5x – 2y = 4
2x + 3y = 13
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
x-value: _______
y-value: _______
Solution: ( , )
Special Cases of Linear Systems
If both value are eliminated when a system of linear equations is solved, then the solution sets are determined as
follows:
Case 1: There are infinitely may solutions if the resulting statement is true.
Case 2: There is no solution (  ) if the resulting statement is false.
Example 4: Solve the system.
Example 5: Solve the system.
16. x – 4y = 2
Solution:
4x – 16y = 8
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
17. 2x – 3y = 7
Solution:
–4x + 6y = 14
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
18. 3x + 7y = 4
Solution:
6x + 14y = 3
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
19. 2x + 3y = 1
Solution:
6x + 9y = 3
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
20. 5x + 2y = 1
Solution:
10x + 4y = 2
______________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________