1. Unit 1 – Lesson 3: Systems of Linear Equations
Systems of Equations
Two or more linear equations that have to be solved at the same time.

2. Systems of Equations Let’s take a look at a few examples.
Your task: I Notice, I Wonder.
Discuss and write down some things that you notice about the examples.
Discuss and write down some things that you wonder about the examples.

3. Systems of Equations: Example 1

4. Systems of Equations: Example 2

5. Systems of Equations: Example 3

6. How do we “solve” a system of equations???
By finding the point where two or more equations, intersect. This is called the SOLUTION.
x + y = 6
y = 2x 6 4
Point of intersection 2 1 2
4/21/2017
Geometry Honors

7. “All I do is Solve” video

8. 3 Methods to Solve System of Equations
Graphing
Substitution Method
Elimination Method

9. 3 Possible Solution Types
No Solution
Infinitely Many Solutions

10. 1 Solution
Point of intersection

11. Another type of solution
How would you describe these lines?
Y = 3x + 2
Y = 3x - 4
What do you think the solution (point of intersection) is?
4/21/2017
Geometry Honors

12. No Solution
Parallel Lines
Will not intersect

13. PARALLEL LINES No Solution: when lines of a graph are parallel
since they do not intersect, there is no solution
Parallel lines have the same slope but different y-intercepts
Slides 10 through 14 show how I explained the different solutions to their worksheet that the students worked on in class. I didn’t focus to much on the solution they found on the worksheet but rather on the type of solution or the concepts they the solutions involved. In each slide I explain each type of solution as well as how a system has these types of solutions. Not focusing to much on the powerpoint I referred back to their worksheet to one of the examples and asked students to remember how we knew the equations graphed were parallel by only looking at the equations. From their prior knowledge students knew that it was because the equations in the system contained the same slope; making connections with new and old material.
4/21/2017
Geometry Honors

14. Another type of solution
What do you notice about the graphs and equations?
y = -3x + 4
3x + y = 4
What do you think the solution (point of intersection) is?
4/21/2017
Geometry Honors

15. Infinitely Many Solutions
SAME LINE

16. INFINITELY MANY SOLUTIONS
Infinite Solutions:
a pair of equations that have the same slope and same y-intercept.
They are the SAME equation (just written in different forms)
Since they are the SAME EQUATION, they have the SAME LINE
Again giving explanations as to how we have such a solution. Just like the last slide, I also questioned students as to how we can find the whether a system has infinite solutions by looking at the equation, triggering prior knowledge.
4/21/2017
Geometry Honors

17. Does it have a solution?
Determine whether the following have one, none, or infinite
solutions by identifying the slope and y-intercept. Explain your reasoning. 1) 2)
2y = 8 - x
y = 2x + 4
y = -6x + 8
y + 6x = 8 3)
x - 5y = 10
-5y = -x +6
4/21/2017
Geometry Honors

18. Does it have a solution?
Determine whether the following have one, none, or infinite
solutions by just looking at the slope and y-intercepts 1) 2) 3)
2y + x = 8
y = 2x + 4
y = -6x + 8
y + 6x = 8
x - 5y = 10
-5y = -x +6
ANS:
One Solution
ANS:
Infinite Solutions
ANS:
No Solution
4/21/2017
Geometry Honors

19. Systems of Equations Video
Systems of Equations: Part 01
Watch carefully as this video explains what a system of equations are and gives a fantastic real-world example of how systems are used in the business world.

20. The Goal of Solving Systems
To find one pair (x, y) of values that satisfies both linear equations.
The one pair of values that makes both equations true.

21. Hamilton High School

22. Hamilton High School 16x + 10y = 240 x + y = 18
What does the x represent?
x: # of outdoor workers
What does the y represent?
y: # of indoor workers

23. Hamilton High School 16x + 10y = 240
What does this equation represent in the problem?
16x + 10y = 240 shows the amount of money that can be earned depending on the # of outdoor and indoor workers

24. Hamilton High School x + y = 18
What does this equation represent in the problem?
x + y = 18 shows that the # of club members who will work

25. Hamilton High School 16x + 10y = 240
Determine three combination of outdoor (x) and indoor (y) workers so that the club earns exactly $240.
SHOW ALL WORK!!

26. Hamilton High School x + y = 18
Do any of the combinations from part d work for the 18 workers that are needed?
SHOW ALL WORK!!

27. Let’s verify
How can we verify that (10,8) is the solution to the system of equation:
16x + 10y = 240
x + y = 18
You must verify the solution into BOTH equations for x AND y.
16(10) + 10(8) = 240
= 240
240 = 240
= 18
18 = 18
Geometry CP

28. Where’s the solution?
Use the graph to estimate a solution for the system of equations (basically what x and y value works for BOTH equations)
SOLUTION: POINT OF INTERSECTION (10, 8)
SOLUTION

29. Hamilton High School
Plugging and chugging numbers is exhausting and very time consuming.
What other strategies could you use to find a pair of values (x,y) that satisfy BOTH equations at the same time?

30. How Do We Graph a Linear Equation???
In order to graph a linear equation it HAS to be in the form y = mx + b, where m is the slope and b is the y-intercept

31. How Do We Graph a Linear Equation???
Let’s Practice: 16x + 10y = 240

32. How Do We Graph a Linear Equation???
Let’s Practice: x + y = 18

33. Let’s Look at the Solution
Complete Problem 1 - Parts a, b, & c.

34. A Better Deal 
When the date for the work project was set, it turned out that only 13 science club members could participate. The club president talked again with the PTA president and got a new pay deal - $20 per outdoor worker and $15 per indoor worker.

35. MATCHING ACTIVITY

36. Verifying Solutions
Determine whether the point (3,8) is a solution to each system of equations.
y = -x – 5
y = x + 5
4x – y = - 4
3x – 2y = 7
2x + y = 14
x + y = 11
4/21/2017
Geometry Honors

37. Verifying Solutions
Determine whether the point (3,8) is a solution to each system of equations.
y = -x – 5
y = x + 5
4x – y = - 4
3x – 2y = 7
2x + y = 14
x + y = 11
2(3) + 8 = 14 (YES)
3 + 8 = 11 (YES)
YES! Both equations are true
8 = -3 – 5 (NO)
8 = (YES)
NO! Only 1 equation is true
4(3) - 8 = - 4 (NO)
3(3) -2(8) = 7 (NO)
NO! Neither equations are true
4/21/2017
Geometry Honors

38. REMIND YOUR SHOULDER BUDDY
How do I know if my answer is correct?
What do you do if the equation is not in y= form?
4/21/2017
Geometry Honors

39. REMIND YOUR SHOULDER BUDDY
How do I know if my answer is correct?
Always replace x and y for BOTH equations to verify your solution.
What do you do if the equation is not in y= form?
You have to rewrite it solving for y so that it can be graphed.
4/21/2017
Geometry Honors

40. GRAPHING CALCULATOR Rewrite equation in y = form
Use the INTERSECT function to find the intersection point
4/21/2017
Geometry Honors

41. GRAPHING CALCULATOR EXAMPLES
y = - 3x and 4x + y = 2
x + y = 1 and 2x + y = 4
3x + y = 1 and –x + 2y = 16
2x + y = 1 and 5x + 4y = 10
4/21/2017
Geometry Honors

42. GRAPHING CALCULATOR EXAMPLES
(2, - 6)
y = - 3x and 4x + y = 2
x + y = 1 and 2x + y = 4
3x + y = 1 and –x + 2y = 16
2x + y = 1 and 5x + 4y = 10
(3, -2)
(-2, 7)
(-2, 5)
4/21/2017
Geometry Honors

43. GRAPHING CALCULATOR EXAMPLES
y = 3x + 2 and y = 3x – 4
y = -3x + 4 and 3x + y = 4
4/21/2017
Geometry Honors

44. GRAPHING CALCULATOR EXAMPLES
Infinitely many
y = 3x + 2 and y = 3x – 4
y = -3x + 4 and 3x + y = 4
No solution
4/21/2017
Geometry Honors

45. Daily Homework Quiz
For use after Lesson 7.1
Use the graph to solve the linear system 1.
3x – y = 5
–x + 3y = 5
ANSWER
(2, 1)

46. Daily Homework Quiz
For use after Lesson 7.1
2. Solve the linear system by graphing.
2x + y = –3
–6x + 3y = 3
ANSWER
(–1, –1)

47. Summarize 3 – 2 – 1 3 methods to solve systems of equations
2 important items to identify when graphing a linear equation
1 way to identify the solution of a graphed systems of equations