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

Good morning everybody . we will take you on a fun learning today. Mrs. Panchaporn Kantayasakun. New generation S.W.K. Mrs. Khrongsri Nampoon. By :. Mrs. Jarin Promsri. System of Linear Equations.

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

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

  2. Mrs. PanchapornKantayasakun New generation S.W.K Mrs. KhrongsriNampoon By: Mrs. JarinPromsri

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

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

  5. 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

  6. Ways to Solve Linear Systems By: Substitution

  7. 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 Solving Linear Systems by Substitution

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

  9. 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

  10. 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)

  11. Solving Linear Systems by Linear Combinations

  12. 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

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

  14. 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

  15. 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 ( )

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

  17. 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

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

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

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

  21. Step 3 Click the plus sign.

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

  23. Step 5 will have a graph

  24. Step 6 Click the plus sign. And Type the equation in the form y=-2-2xand click Done

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

  26. 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

  27. 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

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

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

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

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

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

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

  34. The End

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