1. 8.3 Solving Systems of Linear Equations by Elimination

2. Solve linear systems by elimination.
Objective 1
Solve linear systems by elimination.
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3. If A = B and C = D, then A + C = B + D.
Solve linear systems by elimination.
An algebraic method that depends on the addition property of equality can also be used to solve systems. Adding the same quantity to each side of an equation results in equal sums: If A = B, then A + C = B + C.
We can take this addition a step further. Adding equal quantities, rather than the same quantity, to each side of an equation also results in equal sums:
If A = B and C = D, then A + C = B + D.
Using the addition property to solve systems is called the elimination method. With this method, the idea is to eliminate one of the variables. To do this, one pair of variable terms in the two equations must have coefficients that are opposite.
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4. Using the Elimination Method
CLASSROOM EXAMPLE 1
Using the Elimination Method
Solve the system.
Solution:
The solution set is
A system is not completely solved until values for both x and y are found. Do not stop after finding the value of only one variable. Remember to write the solution set as a set containing an ordered pair
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5. Solving a Linear System by Elimination
Step 1: Write both equations in standard form, Ax + By = C.
Step 2: Transform the equations as needed so that the coefficients of one pair of variable terms are opposites Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either the x- or y-term is 0.
Step 3: Add the new equations to eliminate a variable. The sum should be an equation with just one variable.
Step 4: Solve the equation from Step 3 for the remaining variable.
Step 5: Substitute the result from Step 4 into either of the original equations, and solve for the other variable.
Step 6: Check the solution in both of the original equations Then write the solution set.
It does not matter which variable is eliminated first. Choose the one that is more convenient to work with.
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6. Using the Elimination Method
CLASSROOM EXAMPLE 2
Using the Elimination Method
Solve the system.
Solution:
The solution set is
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7. Multiply when using the elimination method.
Objective 2
Multiply when using the elimination method.
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8. Multiply when using the elimination method.
Sometimes we need to multiply each side of one or both equations in a system by some number before adding will eliminate a variable.
When using the elimination method, remember to multiply both sides of an equation by the same nonzero number.
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9. Using the Elimination Method
CLASSROOM EXAMPLE 3
Using the Elimination Method
Solve the system.
Solution:
The solution set is
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10. Use an alternative method to find the second value in a solution.
Objective 3
Use an alternative method to find the second value in a solution.
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11. Use an alternative method to find the second value in a solution.
Sometimes it is easier to find the value of the second variable in a solution by using the elimination method twice.
When using the elimination method, remember to multiply both sides of an equation by the same nonzero number.
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12. + + CLASSROOM EXAMPLE 4 Solve the system. Solution:
Finding the Second Value by Using an Alternative Method
Solve the system.
Solution: + +
The solution set is
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13. Solve special systems by elimination.
Objective 4
Solve special systems by elimination.
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14. + + CLASSROOM EXAMPLE 5 Solve each system by the elimination method.
Solving Special Systems Using the Elimination Method
Solve each system by the elimination method.
Solution: + +
The solution set is
The solution set is
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