1. Solving Linear Systems Using Linear Combinations There are two methods of solving a system of equations algebraically: Elimination (Linear Combinations) - an equation resulting from the sum of two equations (or multiples of the equations) - used when no variable in either equation has a coefficient of 1 or -1 Substitution

2. Linear Combinations Solving a linear system by linear combinations (addition only): 1.Write the equations so the like terms are lined up in columns. 2.Add the equations. Combining like terms will eliminate one variable. Solve for the remaining variable. 3.Substitute the value from step 2 into either of the original equations and solve for the other variable. 4.Check the solution.

3. Linear Combinations Solve the system: 4x - 8y = 32 16x + 8y = 48 20x = 80Add Like Terms x = 4Divide 4x - 8y = 32Original Equation 4(4) - 8y = 32Substitute 16 - 8y = 32Multiply -8y = 16Subtract 16 from both sides y = -2Divide Answer (4, -2)

4. Check 4x - 8y = 3216x + 8y = 48 4(4) - 8(-2) = 3216(4) + 8(-2) = 48 16 + 16 = 3264 + (-16) = 48 32 = 3264 - 16 = 48 48 = 48

5. Guided Practice Solve the system using linear combinations. 3x + 7y = 84 5x - 7y = 12

6. Arranging Like Terms in Columns Arrange the like terms in columns. Then solve the system. 2x = 12 + 8y 4x + 8y = 24 2x - 8y = 12Re-write the equation to line up 4x + 8y = 24like terms 6x = 36Add like terms x = 6Divide

7. Arranging Like Terms in Columns 4x + 8y = 24 4(6) + 8y = 24Substitute 24 + 8y = 24Multiply 8y = 0Subtract 24 from both sides y = 0Divide Answer (6, 0) Check 2(6) = 12 + 8(0)4(6) + 8(0) = 24 12 = 12 = 024 + 0 = 24 12 = 1224 = 24

8. Guided Practice Solve the system using linear combinations. 2x - 3y = 18 6y - 2x = 24

9. Independent Practice Solve the system using linear combinations. 1.12x + 5y = 90 -4x - 5y = -50 2.3x = 9 + 7y -5y = 3x + 18

10. Linear Combinations Solving a linear system by linear combinations (multiply first): 1.Write the equations so the like terms are lined up in columns. 2.Multiply one equation (or both) by a multiple so that one of the variables has opposite coefficients in the two equations. (Find LCM that is opposites.) 3.Add the equations. Combining like terms will eliminate one variable. Solve for the remaining variable. 4.Substitute the value from step 2 into either of the original equations and solve for the other variable. 5.Check the solution.

11. Linear Combinations Solve the system: 4x - 5y = 29 7x + 10y = 32 2(4x - 5y = 29)Multiply equation by 2; LCM = 10 7x + 10y = 32 8x - 10y = 58Write new equations 7x + 10y = 32 15x = 90Add like terms x = 6Divide

12. Linear Combinations Solve the system: 4x - 5y = 29 7x + 10y = 32 7(6) + 10y = 32Substitute 42 + 10y = 32Multiply 10y = -10Subtract 42 from both sides y = -1Divide Solution (6, -1)

13. Check 4x - 5y = 297x + 10y = 32 4(6) - 5(-1) = 297(6) + 10(-1) = 32 24 + 5 = 2942 + (-10) = 32 29 = 2942 - 10 = 32 32 = 32

14. Guided Practice Solve the system using linear combinations. 4x + 3y = -1 5x + 4y = 1

15. Writing and Using a Linear System You drive 210 miles to a relative's house. It takes you 4 hours. Part of the time you are on a freeway, where the speed limit is 60 mph. The rest of the time you are on smaller roads, where the speed limit is 30 mph. Supposing you drove exactly at the speed limit the whole way, how much time did you spend on each type of road? Use linear combinations to solve. *Think how distance, rate, and time are related d = rt

16. Writing and Using a Linear System (Freeway Time) + (Road Time) = Total Time (Freeway Speed)(Time) + (Road Speed)(Time) = Distance Distance = 210 miles Freeway Time = x Road Time = y Freeway Speed = 60 Road Speed = 30

17. Writing and Using a Linear System x + y = 4Write Equations 60x + 30y = 210 -30(x + y = 4)Multiply to by LCM of a variable 60x + 30y = 210 -30x - 30y = -120 60x + 30y = 210 30x = 90Add the two equation x = 3Divide

18. Writing and Using a Linear System x + y = 4Choose Equation x = 3 3 + y = 4Substitute y = 1Subtract 3 from both sides Answer You spent 3 hours on the freeway and 1 hour on the road. Check your answer.

19. Independent Practice Solve the system using linear combinations. 1.-2x + y = 2 4x - 4y = 4 2.3y = 8 - 7x 8x - 18 = y