1. Good Morning, We are moving on to chapter 3. If there is time today I will show you your test score you can not have them back as I still have several folks that have not finished Quiz Tuesday (9/12) on solving algebraically Chapter 3 homework will be due 9/19

2. SYSTEMS OF LINEAR EQUATIONS Solving Linear Systems Algebraically

3. Solving Systems of Equations Algebraically 1.When you graph, sometimes you cannot find the exact point of intersection. We can use algebra to find the exact point. 2.Also, we do not need to put every equation in slope-intercept form in order to determine if the lines are parallel or the same line. Algebraic methods will give us the same information.

4. Methods of Solving Systems Algebraically We will look at TWO methods to solve systems algebraically: 1) Substitution 2) Elimination

5. Method 1: Substitution Steps: 1.Choose one of the two equations and isolate one of the variables. 2.Substitute the new expression into the other equation for the variable. 3.Solve for the remaining variable. 4.Substitute the solution into the other equation to get the solution to the second variable.

6. Example #1: y = 4x 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 Step 3: Simplify and solve the equation. 7x = -21 x = -3

7. y = 4x 3x + y = -21 Step 4: Substitute back into either original equation to find the value of the other variable. 3x + y = -21 3(-3) + y = -21 -9 + y = -21 y = -12 Solution to the system is (-3, -12).

8. y = 4x 3x + y = -21 Step 5: Check the solution in both equations. y = 4x -12 = 4(-3) -12 = -12 3x + y = -21 3(-3) + (-12) = -21 -9 + (-12) = -21 -21= -21 Solution to the system is (-3,-12).

9. Example #2: x + y = 10 5x – y = 2 Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5x - y = 2 5x -(-x +10) = 2

10. x + y = 10 5x – y = 2 5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x = 2 Step 3: Simplify and solve the equation.

11. x + y = 10 5x – y = 2 Step 4: Substitute back into either original equation to find the value of the other variable. x + y = 10 2 + y = 10 y = 8 Solution to the system is (2,8).

12. x + y = 10 5x – y = 2 Step 5: Check the solution in both equations. x + y =10 2 + 8 =10 10 =10 5x – y = 2 5(2) - (8) = 2 10 – 8 = 2 2 = 2 Solution to the system is (2, 8).

13. Example # 3 Substitution Example: Equation ‘a’:3x + 4y = - 4 Equation ‘b’: x + 2y = 2 Isolate the ‘x’ in equation ‘b’: x = - 2y + 2

14. Example, continued: Equation ‘a’:3x + 4y = - 4 Equation ‘b’:x + 2y = 2 Substitute the new expression, x = - 2y + 2 for x into equation ‘a’: 3(- 2y + 2) + 4y = - 4

15. Example, continued: Equation ‘a’:3x + 4y = - 4 Equation ‘b’:x + 2y = 2 Solve the new equation: 3(- 2y + 2) + 4y = - 4 - 6y + 6 + 4y = - 4 - 2y + 6 = - 4 - 2y = - 10 y = 5

16. Example, continued: Equation ‘a’:3x + 4y = - 4 Equation ‘b’:x + 2y = 2 Substitute y = 5 into either equation ‘a’ or ‘b’: x + 2 (5) = 2 x + 10 = 2 x = - 8 The solution is (-8, 5).

17. Solve by substitution: 1. 2.

18. Method 2: Elimination Steps: 1.Line up the two equations using standard form (Ax + By = C). 2.GOAL: The coefficients of the same variable in both equations should have the same value but opposite signs. 3.If this doesn’t exist, multiply one or both of the equations by a number that will make the same variable coefficients opposite values.

19. Method 2: Elimination Steps, continued: 4.Add the two equations (like terms). 5.The variable with opposite coefficients should be eliminated. 6.Solve for the remaining variable. 7.Substitute that solution into either of the two equations to solve for the other variable.

20. Solving Systems of Equations using Elimination Steps: 1. Place both equations in Standard Form, Ax + By = C. 2. Determine which variable to eliminate with Addition or Subtraction. 3. Solve for the variable left. 4. Go back and use the found variable in step 3 to find second variable. 5. Check the solution in both equations of the system.

21. EXAMPLE #1: STEP 2:Use subtraction to eliminate 5x. 5x + 3y =11 5x + 3y = 11 -(5x - 2y =1) -5x + 2y = -1 5x + 3y = 11 5x = 2y + 1 Note: the (-) is distributed. STEP 3:Solve for the variable. 5x + 3y =11 -5x + 2y = -1 5y =10 y = 2 STEP1: Write both equations in Ax + By = C form. 5x + 3y =11 5x - 2y =1

22. STEP 4: Solve for the other variable by substituting into either equation. 5x + 3y =11 5x + 3(2) =11 5x + 6 =11 5x = 5 x = 1 5x + 3y = 11 5x = 2y + 1 The solution to the system is (1,2).

23. 5x + 3y= 11 5x = 2y + 1 Step 5:Check the solution in both equations. 5x + 3y = 11 5(1) + 3(2) =11 5 + 6 =11 11=11 5x = 2y + 1 5(1) = 2(2) + 1 5 = 4 + 1 5=5 The solution to the system is (1,2).

24. Solving Systems of Equations using Elimination Steps: 1. Place both equations in Standard Form, Ax + By = C. 2. Determine which variable to eliminate with Addition or Subtraction. 3. Solve for the remaining variable. 4. Go back and use the variable found in step 3 to find the second variable. 5. Check the solution in both equations of the system.

25. Example #2: x + y = 10 5x – y = 2 Step 1: The equations are already in standard form:x + y = 10 5x – y = 2 Step 2: Adding the equations will eliminate y. x + y = 10 +(5x – y = 2)+5x – y = +2 Step 3:Solve for the variable. x + y = 10 +5x – y = +2 6x = 12 x = 2

26. x + y = 10 5x – y = 2 Step 4: Solve for the other variable by substituting into either equation. x + y = 10 2 + y = 10 y = 8 Solution to the system is (2,8).

27. x + y = 10 5x – y = 2 x + y =10 2 + 8 =10 10=10 5x – y =2 5(2) - (8) =2 10 – 8 =2 2=2 Step 5: Check the solution in both equations. Solution to the system is (2,8).

28. Method 2: Elimination Example: Equation ‘a’:2x - 4y = 13 Equation ‘b’:4x - 5y = 8 Multiply equation ‘a’ by –2 to eliminate the x’s: Equation ‘a’:-2(2x - 4y = 13) Equation ‘b’:4x - 5y = 8

29. Method 2: Elimination Example, continued: Equation ‘a’:-2(2x - 4y = 13) ------> -4x + 8y = -26 Equation ‘b’:4x - 5y = 8 ------> 4x - 5y = 8 Add the equations (the x’s are eliminated): -4x + 8y = -26 4x - 5y = 8 3y = -18 y = -6

30. Method 2: Elimination Example, continued: Equation ‘a’:-2(2x - 4y = 13) ------> -4x + 8y = -26 Equation ‘b’:4x - 5y = 8 ------> 4x - 5y = 8 Substitute y = -6 into either equation: Solution: (, -6) 4x - 5(-6) = 8 4x + 30 = 8 4x = -22 x =

31. Method 2: Elimination Example 2: Equation ‘a’:-9x + 6y = 0 Equation ‘b’:-12x + 8y = 0 Multiply equation ‘a’ by –4 and equation ‘b’ by 3 to eliminate the x’s: Equation ‘a’:- 4(-9x + 6y = 0) Equation ‘b’:3(-12x + 8y = 0)

32. 36x - 24y = 0 -36x + 24y = 0 0 = 0 Method 2: Elimination Example 2, continued: Equation ‘a’:- 4(-9x + 6y = 0) Equation ‘b’:3(-12x + 8y = 0) What does this answer mean? Is it true?

33. 36 x - 24 y = 0 -36 x + 24 y = 0 0 = 0 Method 2: Elimination Example 2, continued: When both variables are eliminated, if the statement is TRUE (like 0 = 0), then they are the same lines and there are infinite solutions. if the statement is FALSE (like 0 = 1), then they are parallel lines and there is no solution.

34. 36x - 24y = 0 -36x + 24y = 0 0 = 0 Method 2: Elimination Example 2, continued: Since 0 = 0 is TRUE, there are infinite solutions.

35. NOW solve these using elimination: 1.2. 2x + 4y =1 x - 4y =5 2x – y =6 x + y = 3

36. Solving Systems of Three Equations Algebraically 1.When we have three equations in a system, we can use the same two methods to solve them algebraically as with two equations. 2.Whether you use substitution or elimination, you should begin by numbering the equations!

37. Solving Systems of Three Equations Substitution Method 1.Choose one of the three equations and isolate one of the variables. 2.Substitute the new expression into each of the other two equations. 3.These two equations now have the same two variables. Solve this 2 x 2 system as before. 4.Find the third variable by substituting the two known values into any equation.

38. Solving Systems of Three Equations Linear Combination Method 1.Choose two of the equations and eliminate one variable as before. 2.Now choose one of the equations from step 1 and the other equation you didn’t use and eliminate the same variable. 3.You should now have two equations (one from step 1 and one from step 2) that you can solve by elimination. 4.Find the third variable by substituting the two known values into any equation.