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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Solving Systems of Linear Equations by Substitution
4.2
Solving Systems of Linear Equations by Substitution 1
Solve linear systems by substitution.
Solve special systems by substitution.
Solve linear systems with fractions. 2 3
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3. Solve linear systems by substitution.
Objective 1
Solve linear systems by substitution.
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4. Solve linear systems by substitution.
Graphing to solve a system of equations has a serious drawback. It is difficult to find an accurate solution, such as
, from a graph. One algebraic method for solving a system of equations is the substitution method.
This method is particularly useful for solving systems in which one equation is already solved, or can be solved quickly, for one of the variables.
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5. Solve linear systems by substitution. (cont’d)
To solve a system by substitution, follow these steps.
Step 1: Solve one equation for either variable. If one of the variables has coefficient 1 or −1, choose it, since it usually makes the substitution method easier.
Step 2: Substitute for that variable in the other equation. The result should be an equation with just one variable.
Step 3: Solve the equation from Step 2.
Step 4: Substitute the result from Step 3 into the equation from Step 1 to find the value of the other variable.
Step 5: Check the solution in both of the original equations Then write the solution set.
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6. EXAMPLE 1 Solve the system by the substitution method. Solution:
Using the Substitution Method
Solve the system by the substitution method.
Solution:
The solution set found by the substitution method will be the same as the solution found by graphing. The solution set is the same; only the method is different.
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7. EXAMPLE 2 Solve the system by the substitution method. Solution:
Using the Substitution Method
Solve the system by the substitution method.
Solution:
Be careful when you write the ordered-pair solution of a system. Even though we found y first, the x-coordinate is always written first in the ordered pair.
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8. EXAMPLE 3 Use substitution to solve the system. Solution:
Using the Substitution Method
Use substitution to solve the system.
Solution:
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9. Solve special systems by substitution.
Objective 2
Solve special systems by substitution.
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10. EXAMPLE 4 Use substitution to solve the system. Solution:
Solving an Inconsistent System by Substitution
Use substitution to solve the system.
Solution:
Since the statement is false, the solution set is Ø.
It is a common error to give “false” as the solution of an inconsistent system. The correct response is Ø.
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11. EXAMPLE 5 Solve the system by the substitution method. Solution:
Solving a System with Dependent Equations by Substitution
EXAMPLE 5
Solve the system by the substitution method.
Solution:
Since the statement is true every solution of one equations is also a solution to the other, so the system has an infinite number of solutions and the solution set is {(x,y)|x + 3y = −7}.
It is a common error to give “true” as the solution of a system of dependent equations. Remember to give the solution set in set-builder notation using the equation in the system that is in standard form with integer coefficients that have no common factor (except 1).
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12. Solve linear systems with fractions.
Objective 3
Solve linear systems with fractions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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13. EXAMPLE 6 Solve the system by the substitution method. Solution:
Using the Substitution Method with Fractions as Coefficients
EXAMPLE 6
Solve the system by the substitution method.
Solution:
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