1. Systems of Equations in Two Variables
8.1
Systems of Equations in Two Variables
Translating
Identifying Solutions
Solving Systems Graphically
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2. System of Equations
A system of equations is a set of two or more equations, in two or more variables, for which a common solution is sought.
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3. Example
T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each. In all, $ was taken in for the shirts. How many of each kind were sold? Set up the equations but do not solve.
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4. Identifying Solutions
A solution of a system of two equations in two variables is an ordered pair of numbers that makes both equations true.
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5. Example Determine whether (1, 5) is a solution of the system
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6. Solving Systems Graphically
One way to solve a system of two equations is to graph both equations and identify any points of intersection. The coordinates of each point of intersection represent a solution of that system.
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7. Example Solve the system graphically. x – y = 1 x + y = 5 (3, 2)
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8. Example Solve the system graphically.
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9. Example Solution Solve the system graphically. Graph both equations.
The same line is drawn twice. Any solution of one equation is a solution of the other. There is an infinite number of solutions. The solution set is
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10. When we graph a system of two linear equations in two variables, one of the following three outcomes will occur.
The lines have one point in common, and that point is the only solution of the system. Any system that has at least one solution is said to be consistent.
The lines are parallel, with no point in common, and the system has no solution. This type of system is called inconsistent.
The lines coincide, sharing the same graph. This type of system has an infinite number of solutions and is also said to be consistent.
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11. When one equation in a system can be obtained by multiplying both sides of another equation by a constant, the two equations are said to be dependent. If two equations are not dependent, they are said to be independent.
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12. Solving by Substitution or Elimination
8.2
The Substitution Method
The Elimination Method
Solving by Substitution or Elimination
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13. The Substitution Method
Algebraic (nongraphical) methods for solving systems are often superior to graphing, especially when fractions are involved. One algebraic method, the substitution method, relies on having a variable isolated.
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14. Example Solution Solve the system (1) (2)
The equations are numbered for reference.
Solution
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15. Example Solution Solve the system (1) (2)
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16. Example Solve the system Solution (1) (2)
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17. The Elimination Method
The elimination method for solving systems of equations makes use of the addition principle: If a = b, then a + c = b + c.
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18. Example Solve the system (1) (2)
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19. Example Solve the system (1) (2)
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20. Example Solve the system Solution (1) (2)
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21. Rules for Special Cases
When solving a system of two linear equations in two variables:
1. If we obtain an identity such as 0 = 0, then the system has an infinite number of solutions. The equations are dependent and, since a solution exists, the system is consistent.
2. If we obtain a contradiction such as 0 = 7, then the system has no solution. The system is inconsistent.
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22. Solving Applications: Systems of Two Equations
8.3
Solving Applications: Systems of Two Equations
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23. Total-Value Problems
The next example involves two types of items, the quantity of each type bought, and the total value of the items. We refer to this type of problem as a total-value problem.
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24. Example
T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each. In all, $ was taken in for the shirts. How many of each kind were sold?
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25. Mixture Problems
The next example is similar to the last example. Note that in each case, one of the equations in the system is a simple sum while the other equation represents a sum of products. We refer to this type of problem as a mixture problem.
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26. Problem-Solving Tip
When solving a problem, see if it is patterned or modeled after a problem that you have already solved.
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27. Example
An employee at a small cleaning company wishes to mix a cleaner that is 30% acid and another cleaner that is 50% acid. How many liters of each should be mixed to get 20 L of a solution that is 35% acid?
Solution +
30% acid % acid % acid =
t liters
f liters
20 liters
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28. Motion Problems
The next example deals with distance, speed (rate), and time. We refer to this type of problem as a motion problem.
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29. Distance, Rate, and Time Equations
If r represents rate, t represents time, and d represents distance, then
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30. Example
Alex’s motorboat took 4 hr to make a trip downstream with a 5-mph current. The return trip against the same current took 6 hr. Find the speed of the boat in still water.
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31. Tips for Solving Motion Problems
1. Draw a diagram using an arrow or arrows to represent distance and the direction of each object in motion.
2. Organize the information in a chart.
3. Look for times, distances, or rates that are the same. These often can lead to an equation.
4. Translating to a system of equations allows for the use of two variables.
5. Always make sure that you have answered the question asked.
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