2. Systems of Linear Equations
Chapter 5
Systems of Linear Equations

3. Systems of Linear Equations In Two Variables
5.1
Systems of Linear Equations
In Two Variables

4. 5.1 Systems of Linear Equations in Two Variables
Objectives
Solve linear systems by graphing.
Decide whether an ordered pair is a solution of a linear system.
Solve linear systems (with two equations and two variables) by substitution.
Solve linear systems (with two equations and two variables) by elimination.
Solve special systems.
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5. 5.1 Systems of Linear Equations in Two Variables
Systems of Equations
Suppose we wished to cut a 10-foot length of conduit into two pieces such that one piece is 4 feet longer than the other. We are looking for two numbers whose sum is 10 and whose difference is 4. If we let x represent the larger of the two numbers and y the other number, we immediately get two equations.
We call such an arrangement a system of two equations in two variables.
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6. 5.1 Systems of Linear Equations in Two Variables
Solution of a Systems of Equations
A solution of a system of equations is an ordered pair that satisfies both equations at the same time. Is there such an ordered pair? Is there more then one such pair? To answer these questions, we can look at the graph of these two equations on the same coordinate system. 12 8 4 10
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7. 5.1 Systems of Linear Equations in Two Variables
Solution of a Systems of Equations
To be sure that (7, 3) is a solution of both equations, we can check by substituting 7 for x and 3 for y in both equations. 12 8 4 10
True
True
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8. 5.1 Systems of Linear Equations in Two Variables
Solution of a Systems of Equations
Given the following system of equations, determine whether the given ordered pair is a solution of the system.
The ordered pair must be a solution of both equations to be a solution.
A solution of the system.
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9. 5.1 Systems of Linear Equations in Two Variables
Deciding Whether an ordered Pair is a Solution
Given the following system of equations, determine whether the given ordered pair is a solution of the system.
Since the ordered pair is not a solution of the first equation, we need not check further; the ordered pair is not a solution.
Not a solution of the system.
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10. 5.1 Systems of Linear Equations in Two Variables
Possible Solutions for a Linear System of Two Variables
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11. 5.1 Systems of Linear Equations in Two Variables
The Substitution Method for Solving Systems
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12. 5.1 Systems of Linear Equations in Two Variables
Solving a System by Substitution
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13. 5.1 Systems of Linear Equations in Two Variables
Solving a System by Substitution
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14. 5.1 Systems of Linear Equations in Two Variables
Solving a System by Substitution
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15. 5.1 Systems of Linear Equations in Two Variables
The Elimination Method for Solving Systems
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16. 5.1 Systems of Linear Equations in Two Variables
Solving a System by Elimination
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17. 5.1 Systems of Linear Equations in Two Variables
Solving a System by Elimination
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18. 5.1 Systems of Linear Equations in Two Variables
Solving a System by Elimination
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19. 5.1 Systems of Linear Equations in Two Variables
Solving a System by Elimination
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20. 5.1 Systems of Linear Equations in Two Variables
Solving a System with Fractional Coefficients
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21. 5.1 Systems of Linear Equations in Two Variables
Solving a System with Fractional Coefficients
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22. 5.1 Systems of Linear Equations in Two Variables
Some Special Systems
Some systems of linear equations have no solution or an infinite number of solutions.
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23. 5.1 Systems of Linear Equations in Two Variables
Some Special Systems
We could get equation (1) by multiplying equation (2) by 3. Because of this, equations (1) and (2) are equivalent and have the same graph.
The solution set is the set of all (infinite number of) points on the line with equation 3x + 4y = 6, written
{(x, y) | 3x + 4y = 6} and read “the set of all ordered pairs (x, y) such that
3x + 4y = 6.”
This is a dependent system of equations.
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24. 5.1 Systems of Linear Equations in Two Variables
Some Special Systems
The system is dependent; the lines are the same.
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25. 5.1 Systems of Linear Equations in Two Variables
Some Special Systems
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26. 5.1 Systems of Linear Equations in Two Variables
Some Special Systems
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27. 5.1 Systems of Linear Equations in Two Variables
Some Special Systems
The system is inconsistent; the lines have no points in common.
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