1. 6.3 Using Elimination to Solve Systems

2. Adding Left Sides and Right Sides of Two Equations
If a = b and c = d, then
a + c = b + d
In words, the sum of the left sides of two equations is equal to the sum of the right sides.

3. Example: Solving a System by Elimination
Solve the system
Equation (1)
Equation (2)

4. Solution
We begin by adding the left sides and adding the right sides of the two equations:
Equation (1)
Equation (2)

5. Solution
By “eliminating” the variable x, we now have an equation in one variable. Next, we solve that equation for y:

6. Solution
Then, we substitute 3 for y in either of the original equations and solve for x:
Equation (1)

7. Solution
The solution is (1, 3). We check that (1, 3) satisfies both equations (1) and (2):

8. Using Elimination to Solve a Linear System
To use elimination to solve a system of two linear equations,
1. Use the multiplication property of equality (Section 4.3) to get the coefficients of one variable to be equal in absolute value and opposite in sign.
2. Add the left sides and add the right sides of the equations to eliminate one of the variables.

9. Using Elimination to Solve a Linear System
3. Solve the equation in one variable found in step 2.
4. Substitute the solution found in step 3 into one of the original equations, and solve for the other variable.

10. Example: Solving a System by Elimination
Solve the system
Equation (1)
Equation (2)

11. Solution
To eliminate the y terms, we multiply both sides of equation (1) by 3 and multiply both sides of equation (2) by 4, yielding the system

12. Solution
The coefficients of the y terms are now equal in absolute value and opposite in sign. Next, we add the left sides and add the right sides of the equations and solve for x:

13. Solution Substituting 1 for x in equation (1) gives Equation (1)
The solution is (1, –2).

14. Solving Inconsistent Systems and Dependent Systems by Elimination
If the result of applying elimination to a linear system of two equations is
a false statement, then the system is inconsistent; that is, the solution set is the empty set.
a true statement that can be put into the form a = a, then the system is dependent; that is, the solution set is the set of ordered pairs represented by every point on the (same) line.

15. Example: Solving a System by Elimination
Solve the system
Equation (1)
Equation (2)

16. Solution
To eliminate the y terms, we multiply both sides of equation (1) by –3, yielding the system

17. Solution
Now that the coefficients of the x terms (and those of the y terms) are equal in absolute value and opposite in sign, we add the left sides and add the right sides of the equations:

18. Solution
Since 0 = 0 is a true statement of the form a = a, we conclude that the system is dependent and that the solution set of the system is the set of ordered pairs represented by the points on the line 2x – 7y = 5 and the (same) line 6x – 21y = 15.

19. Solve a System of Two Linear Equations
Any linear system of two equations can be solved by graphing, substitution, or elimination. All three methods will give the same result.