1. Systems of linear equations

2. Warm up
-Take 10 minutes and with your partner next to you discover the price of the chocolate chip muffin, cup of coffee and cinnamon roll.
$2.25
$4.55
$3.75

3. Introduction to system of linear equations
A system of linear equations consists of 2 or more linear equations that use the same variables.
Ex: 2x+3y=10 OR 5a+2b=14
4x-7y=7 a+6b=14

4. The solution to a system of equation is the point or points that make both or all of the equations true.
Remember that a point is represented by an ordered pair (x,y)

5. Let’s look at an example- is the given ordered pair a solution to the given system?
3𝑥+7𝑦=12 7𝑥−𝑦=− (−3,3)
2𝑥−7=−𝑦 −5𝑥+13=𝑦 (2,3) No
YES

6. There are 3 ways you can solve for a system of linear equations:
Graphing
Substitution
Elimination

7. Infinitely many solutions
When solving for a system of linear equations you can have 3 different types of solutions:
One solution
No solution
Infinitely many solutions

8. To solve a system of linear equation by graphing I first need to
Convert to slope intercept form
State m and b for both equations
Find point of intersection
State the answer as an ordered pair

9. 2𝑥−2𝑦=−8 2𝑥+2𝑦=4

10. 2𝑥−2𝑦=−8 2𝑥+2𝑦=4 One solution: (-1,3)

11. 𝑦=−2𝑥+5 𝑦=−2𝑥+1

12. 𝑦=−2𝑥+5 𝑦=−2𝑥+1 No solution

13. 𝑥+𝑦=−2 2𝑥−3𝑦=−9

14. 𝑥+𝑦=−2 2𝑥−3𝑦=−9 One solution (-3,1)

15. 2𝑥+𝑦=−2 2𝑥+2=−𝑦

16. 2𝑥+𝑦=−2 2𝑥+2=−𝑦 Infinite solutions

17. Looking at the above graphs and solutions, I can tell that the system will have an infinite number of solutions when the lines have the ___________________________________________. The system will have no solutions when ____________________________________________. The system of equations will have one solution when_____________________________________________.

18. Looking at the above graphs and solutions, I can tell that the system will have an infinite number of solutions when the lines have the same y-intercept b, and the same slope m. The system will have no solutions when the lines have the same slope m, but different y-intercepts b. The systems will have one solution when the lines have different slopes m.

19. Solving Systems by substitution:
Solve one equation for either variable (x= or y=)
Substitute the expression from step one into the second equation
Solve the second equation for the given variable
Plug your solution back into the first equation
Write your solution as an ordered pair
Go back and check your solution in BOTH equations.

20. 2x-3y=-24 X+6y=18 Solve one equation for either variable (x= or y=)
Which equation is the easiest to solve for one variable?

21. 2x-3y=-24 X+6y=18
Solve one equation for either variable (x= or y=)
x+6y=18
-6y y
X=18-6y

22. Substitute the expression from step one into the second equation
2x-3y= X=18-6y
2 (18-6y)-3y=-24

23. Solve the second equation for the given variable
2 (18-6y)-3y=-24
36-12y-3y=-24
36-15y=-24
-15y=-60
Y=4

24. Plug your solution back into the first equation
X+6y=18
Y=4
X+6(4)=18
X+24=18
X=-6

25. Write your solution as an ordered pair
(-6,4)

26. Go back and check your solution in BOTH equations.
2x-3y=-24 X+6y=18 (-6,4)
2(-6)-3(4)= =-24
-24=-24
-6+6(4)=18
-6+24=18
18=18 ✔️ ✔️