1. College Algebra Chapter 5 Systems of Equations and Inequalities
Section 5.1
Systems of Linear Equations in Two Variables and Applications
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2. Concepts
1. Identify Solutions to Systems of Linear Equations in Two Variables 2. Solve Systems of Linear Equations in Two Variables 3. Use Systems of Linear Equations in Applications

3. Concept 1 Identify Solutions to Systems of Linear Equations in Two Variables

4. Example 1
Determine if the ordered pair is a solution to the system. (-3, 5) 2x + 6y = 24 x + 5y = 3

5. Example 2
Determine if the ordered pair is a solution to the system.

6. Skill Practice 1
Determine if the ordered pair is a solution to the system.

7. Identify Solutions to Systems of Linear Equations in Two Variables (1 of 2)
One Unique Solution: A system of linear equations that represents intersecting lines has exactly one solution. No Solution: If a system of linear equations represents parallel lines, then the lines do not intersect, and the system has no solution. In such a case, we say that the system is inconsistent.

8. Identify Solutions to Systems of Linear Equations in Two Variables (2 of 2)
Infinitely Many Solutions: If a system of linear equations represents the same line, then all points on the common line satisfy each equation. Therefore, the system has infinitely many solutions. In such a case we say that the equations are dependent.

9. Concept 2 Solve Systems of Linear Equations in Two Variables

10. Solve Systems of Linear Equations in Two Variables
Substitution:
Isolate one variable.
Substitute into the other equation.
Solve the resulting equation for one variable.
Addition:
Write both equations as Ax + By = C.
Create opposite coefficients for one variable.
Add and solve for one variable.
In either method, substitute known value back into one original equation to find the other value.

11. Example 3 Solve the system by the substitution method. 2x + 3y = 10

12. Skill Practice 2
Solve the system by using the substitution method. 3x + 4y = 5 x - 3y = 6

13. Example 4
Solve the system by the addition method. 2x + 3y = 10 5x – y = -26

14. Skill Practice 3
Solve the system by using the addition method. 2x - 9y = 1 3x = y

15. Example 5
Solve. 5x + 0.3y = x - 0.1y = 0.28

16. Example 6
Solve. 2x + 4y = 4

17. Example 7
Solve. 3(x - 1) + 10 = -y 6(y - 5) = 4x - 28

18. Skill Practice 4
Solve the system by using the addition method. 2(x - 2y) = y + 14

19. Example 8
Solve. 10x - 5y = 30 y = 2x - 6

20. Example 9
Solve.

21. Skill Practice 5
Solve the system. 3x – y = 2 -9x + 3y = 4

22. Skill Practice 6
Solve the system. x = 5 - 3y 2x + 6y = 10

23. Concept 3 Use Systems of Linear Equations in Applications

24. Example 10
David borrowed a total of $10,000 to pay for his final year of graduate school. He borrowed part of the money through his school with a Perkins loan that charged 5% simple interest per year and the remainder of the money from his aunt at a rate of 1.4% simple interest per year. At the end of the year he paid $428 in interest. How much did he borrow from each source?

25. Skill Practice 7
How many ounces of 20% and 30% acid solution should be mixed to produce 15 oz 30% acid solution?

26. Example 11
A plane flies from Atlanta to Los Angeles against the wind in 5 hours. The return trip with the wind takes only 4 hours. If the distance between Atlanta and LA is 3200 kilometers, find the speed of the plane in still air and the speed of the wind.

27. Skill Practice 8
A boat takes 3 hours to go 24 mi upstream against the current. It can go downstream with the current distance of 48 mi in the same amount of time. Determine the speed of the boat in still water and the speed of the current.

28. Skill Practice 9
A storage company rents its units for $120 per month. The company has fixed monthly cost of $2100 and variable costs (air conditioning and service) of $50 per unit.
Write a cost function representing the monthly cost C(x) to the company for x units.
Write a revenue function representing the revenue R(x) when x units per month are rented.
Determine the number of units that must be rented in a month for the company to break even.