3. Linear Systems Systems of Equations Solve by Graphing Solve by Substitution Solve by Elimination Applications of Systems

4. Solutions to Systems of Equations Systems of Equations: Two or more equations working together, simultaneous equations The solution to a system of equations is the point where the lines intersect

5. Finding the Solution (-2,2) Graph System of Equations x – y = -4 x + 2y = 2 Solution (x,y) (-2,2) x – y = -4 x + 2y = 2 (-2) – 2 = -4 (-2) + 2(2) = 2 -4 = -4 -2 + 4 = 2 2 = 2

6. No Solution Graph System of Equations

7. Infinite Solutions Graph System of Equations x 2

8. Solve by Graphing y = 3x + 2 y = -2x + 7 Create a table of values for each equation xy = 3x + 2y(x,y) 0y = 3(0) + 22(0,2) 1y = 3(1) + 25(1,5) xy = -2x + 7y(x,y) 0y = -2(0) + 77(0,7) 1y = -2(1) + 75(1,5) y = -2x + 7 y = 3x + 2

9. Graph the points and connect them to form lines (x,y) (0,2) (1,5) (x,y) (0,7) (1,5) y = 3x + 2 y = -2x + 7 (1,5)

10. Solve by Substitution x + 2y = 10 x + 4y = 8 One equation can be substituted into another equation by replacing a variable

11. Isolate one variable in one equation. Substitute the expression from one equation into the other equation. Solve for the unknown variable. Substitute the value that you just found into one of the original equations to find the other variable. Check your work on both equations using substitution. Step 1: Step 2: Step 3: Step 4: Step 5 : Substitution Method

12. x + 2y = 10 x + 4y = 8 Solve for the variable x in equation 1 x + 2y = 10 x + 2y – 2y = 10 – 2y x = 10 – 2y Equation 1 Equation 2

13. x = ( 10-2y ) x + 4y = 8 (10 – 2y) + 4y = 8 10 – 2y + 4y = 8 10 + 2y = 8 10 – 10 + 2y = 8 – 10 2y = -2 Equation 2 2 y = -1 Equation 1

14. x + 2y = 10 y = -1 x + 2(-1) = 10 x + -2 = 10 x = 12 The solution to the system of equations is (12,-1) Equation 1 y = -1 x + 2y = 10 x + 4y = 8 Equation 1 Equation 2

15. Solve by Elimination -2x + 9y = 27 2x + 6y = -12 Arrange both equations so the corresponding variables are above one another

16. Arrange both equations so the corresponding variables are above one another. You may need to rewrite one of the equations to do this. Look for variables that have opposite coefficients (+/–) or coefficients that are multiples (such as 2 and 4, and so on). If needed, multiply one equation to create coefficients that add to zero (or cancel when added). Add both equations, eliminating one variable. Solve for the remaining variable. Substitute the value you found in Step 5 into one of the original equations. Solve for the second variable. You now have a solution that can be written as an ordered pair, such as (x, y). Check your work using substitution to verify the solution works in both equations. Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Elimination Method

17. Solve by Elimination 2x + 3y = 20 - 2x + y = 4 2x - 2x 2x + 3y = 20 - 2x + y = 4 4y = 24 y = 6 Remember that opposite coefficients cancel so, 2x and -2x result in 0x. When you combine opposite coefficients, the variable cancels out.

18. Solve for the remaining variable x 2x + 3y = 20 y = 6 2x + 3(6) = 20 2x + 18 = 20 2x + 18 – 18 = 20 – 18 2x = 2 x = 1 The solution to the system of equations is (1,6)

19. Applications of Systems Write an equation for each situation Solve the system Substitute the value for one variable to find the other variable

20. Ben is 11 years older than Ava. Ben’s age is 15 years less than three times Ava's age. Ben's age (b) and Ava's age (a): b = 3a − 15 b = a + 11

21. b = 3a − 15 b = a + 11 3a − 15 = a + 11 3a – a = 11 + 15 2a = 26 a = 13 b = 13 + 11 b = 24

22. A school fair ticket costs $8 per adult and $1 per child. The total number of adults and children who went to the fair was 30, and the total money collected was $100. How many children and adults attended the fair? a = adults c = children

23. A school fair ticket costs $8 per adult and $1 per child. The total number of adults and children who went to the fair was 30, and the total money collected was $100. How many children and adults attended the fair? a = adults c = children a + c = 30 8a + 1c = 100

24. (-1) a = adults c = children = 70 a = 10 8a + 1c = 100 a + c = 30 8a + 1c = 100 -1a - 1c = -30 (-1) 7a 20 10 (-1)

25. elimination graphing substitution LINEAR SYSTEMS