1. College Algebra Chapter 5 Systems of Equations and Inequalities
Section 5.1
Systems of Linear Equations in Two Variables and Applications

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. Example 1:
Determine if the ordered pair is a solution to the system.

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

6. Identify Solutions to Systems of Linear Equations in Two Variables
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.
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.

7. 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

8. 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.

9. Example 3:
Solve the system by the substitution method.

11. Example 4:
Solve the system by the addition method.

13. Example 5:
Solve.

14. Example 6:
Solve.

15. Example 7:
Solve.

17. Example 8:
Solve.

18. Example 9:
Solve.

21. 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

22. 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?

23. Example 10 continued:

25. 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.

26. Example 11 continued: