2. Good morning everybody. we will take you on a fun learning today.

3. New generation S.W.K By: Mrs. Jarin Promsri Mrs. Panchaporn Kantayasakun Mrs. Khrongsri Nampoon

4. System of Linear Equations How to: solve by graphing, substitution, linear combinations, and special types of linear systems

5. What is a Linear System, Anyways? A linear system includes two, or more, equations, and each includes two or more variables. linear system.When two equations are used to model a problem, it is called a linear system.

6. Before You Begin…Important Terms to know Linear system: two equations that form one equation Solution: the answer to a system of linear equation; must satisfy both equations ***: a solution is written as an ordered pair: (x,y) Leading Coefficient: any given number that is before any given variable (for example, the leading coefficient in 3x is 3.) Isolate: to get alone

8. Solving Linear Systems by Substitution Basic steps: 1. Solve one equation for one of its variables 2. Substitute that expression into the other equation and solve for the other variable 3. Substitute that value into first equation; solve 4. Check the solution

9. Example: The Substitution Method Here’s the problem: Equation one-x+y=1 Equation two2x+y=-2

10. First, solve equation one for y y = x+1 Next, substitute the above expression in for “y” in equation two, and solve for x Here’s how: Equation two 2x+y=-2 2x+ (x+1)=-2 3x+1=-2 3x=-3 x=-1

11. Congratulations! You now know x has a value of –1…but you still need to find “y”. To do so… First, write down equation one y=x+1 y= (-1)+1 y=0 So, now what? You’re done; simply write out the solution as (-1,0) ***Did you remember? To write a solution, once you’ve found x and y, you must put x first and then y: (x,y)

12. Solving Linear Systems by Linear Combinations

13. Solving Systems by means of Linear Combinations Basic steps: 1. Arrange the equations with like terms in columns 2. After looking at the coefficients of x and y, you need to multiply one or both equations by a number that will give you new coefficients for x or y that are opposites. 3. Add the equations and solve for the unknown variable 4. Substitute the value gotten in step 3 into either of the original equations; solve for other variable 5. Check the solution in both original equations

14. Example: Solving Systems by Linear Combinations Solve this linear equation: Equation One: 3x+5y = 6 Equation Two: -4x+2y =5

15. Solve the linear system Equation 1: 3x+5y=6 Equation 2: -4x+2y=5 3x+5y= 6 --------  -4x+2y= 5 --------  4   ;4(3x+5y)= 4  6 12x+20y= 24 --------  3   ;3(-4x+2y) = 3  5 -12x +6y= 15 --------   +  ; 12x+ 20y -12x + 6y = 24 + 15 26y = 39

16. Equation 2: -4x+2y=5 Substitute the value you just found for -4x+2( ) = 5 -4x+3 = 5 -4x = 2 x = The solution to the example system is ( )

17. A Final way to Solve Systems: Graph and Check

18. Types of Solutions of Systems of Equations One solution – the lines cross at one point No solution – the lines do not cross Infinitely many solutions – the lines coincide

19. An Example of the Quick graph on and Check Method Here’s the problem: Equation one-x+y=1 Equation two2x+y=-2

20. Step 1 Download Application Quick graph from programe App Store.

21. Step 2 open App Quick graph on Iphone or samsung etc.

22. Step 3 Click the plus sign.

23. Step 4 Type the equation in the form y=1+x and click Done.

24. Step 5 will have a graph

25. Step 6 Click the plus sign. And Type the equation in the form y=-2-2x and click Done

26. Step 7 will have a graph for equation y=-2-2x Answer to the equation is the graph intersect (-1,0)

27. Fun, Fun: Exercises 1. Solve the following Linear System Equation one: 3x-4y=10 Equation two: 5x+7y=3 2. Solve the following Linear system Equation one: x-6y=-19 Equation two: 3x-2y=-9 3. Solve the following Linear system Equation one: x+3y=7 Equation two: 4x-7y=-10

28. 4. Use linear combinations to solve this system Equation one: x+2y=5 Equation two: 3x-2y=7 5. Use linear combinations to solve this system Equation one: 3x-5y=-4 Equation two: -9x+7y=8

29. Answers to the Exercises 1. (2,-1) 2. (-1,3) 3. (1,2) 4. (3,1) 5. (-0.5, 0.5)

30. Check out the answers from of the Quick graph 3x-4y=10 5x+7y=3 1.

31. Check out the answers from of the Quick graph 2. 3x-2y=-9 x-6y=-19

32. Check out the answers from of the Quick graph 3. x+3y=7 4x-7y=-10

33. Check out the answers from of the Quick graph 4. x+2y=5 3x-2y=7

34. Check out the answers from of the Quick graph 5. -9x+7y=8 3x-5y=-4