1. Intro to Systems of Equations and Graphing

2. What is a System of Equation?
A SYSTEM OF EQUATIONS is: A SOLUTION to the SYSTEM OF EQUATIONS is: A POINT OF INTERSECTION is:
two or more equations with the same variable being compared.
the point that both equations have in common: THE POINT OF INTERSECTION
the point where both lines meet

3. Write the solution of each system of equation as an ordered pair.
(4, -3)
(8, -3)
Both systems have only ONE SOLUTION.

4. Write the solution of each system of equation as an ordered pair.
(2, -2)
(1, 2)

5. Write the solution of each system of equation as an ordered pair.∅ ∅
Parallel lines do not intersect. Therefore, there is NO SOLUTION!

6. Write the solution of each system of equation as an ordered pair.
What is different about this graph?
There appears to only be one line graphed. We call this coinciding lines.
Because there are more than one line on top of each other, what can we conclude about the solution set?
There are infinite many solutions ℝ

7. Write the solution of each system of equation as an ordered pair.
(-3, 0)

8. Graphing System of Equations
When finding the solution to a system of equation, finding the point of intersection is required. To do this, the equations must be graphed. Therefore, the equation must be put in SLOPE-INTERCEPT FORM.
𝑦=𝑚𝑥+𝑏

9. Find the solution set for
𝒚=− 𝟏 𝟐 𝒙+𝟒 𝟑𝒙 −𝟐𝒚=𝟖
Procedures
𝑦= − 1 2 𝑥+4
3𝑥−2𝑦=8
Put each equation in Slope-
Intercept Form
Identify the slope and y-
intercept
m = ____ b = ____
m = ____ b =____
 3x – 2y = 8
-3x x
-2y = -3x + 8
𝑦= 3 2 𝑥 −4
Already in the correct format!
− 𝟏 𝟐
𝟑 𝟐 4 -4

10. Draw each line on the same grid.
Procedures
𝑦= − 1 2 𝑥+4
3𝑥−2𝑦=8
Identify the slope and y- intercept
m = ____ b = ____
m = ____ b =____
− 𝟏 𝟐
𝟑 𝟐 -4 4
What type of lines are there?
What type of solutions do these lines have?
What is/are the solution(s) located?
3𝑥−2𝑦=8
Intersecting lines
𝑦= − 1 2 𝑥+4
One solution
( 4, 2 )

11. Find the solution set for
𝟐𝒙+𝒚=𝟒 𝒙 −𝒚=𝟐
Procedures
𝟐𝒙+𝒚=𝟒
𝒙 −𝒚=𝟐
Put each equation in Slope-
Intercept Form   
Identify the slope and y-
intercept
m = ____ b = ____
m = ____ b =____
2x + y = 4
- 2x x
y = -2x + 4
x - y = 2
-x x
-y = -x + 2
y = x - 2
-1( ) −𝟐 4 𝟏 -2

12. Draw each line on the same grid.
 Procedures
𝟐𝒙+𝒚=𝟒
𝐱 −𝐲=𝟐
Identify the slope and y- intercept
m = ____ b = ____
m = ____ b =____ −𝟐 4 1 -2
What type of lines are there?
What type of solutions do these lines have?
What is/are the solution(s) located?
x −y=2
𝑦= −2𝑥+4
Intersecting lines
One solution
( 2, 0)

13. Find the solution set for
𝒚=𝟐𝒙 −𝟑 𝟐𝒙 −𝒚=−𝟏
Procedures
𝑦=𝟐𝐱 −𝟑
2𝐱 −𝐲 =−𝟏
Put each equation in Slope-
Intercept Form
Identify the slope and y-
intercept
m = ____ b = ____
m = ____ b =____
 2x – y = -1
-2x x
-y = -2x - 1
𝑦=2𝑥+1
Already in the correct format!
-1( ) 𝟐 -3 𝟐 1

14. Graph each equation in the graphing calculator and answer the following questions.
 Procedures
𝒚=𝟐𝒙 −𝟑
𝟐𝒙−𝒚=−𝟏
Identify the slope and y- intercept
m = ____ b = ____
m = ____ b =____ 𝟐 -3 𝟐 1
What type of lines are there?
What type of solutions are there?
What do you notice about the equations when they are in slope intercept form AND do you think this has an effect on the solution?
Parallel lines
NO SOLUTIONS
When the equations are in slope-intercept form, you notice the slopes are the same. When the slopes are the same, the lines will be parallel and have no solutions.

15. Find the solution set for
𝟑𝒙 −𝒚=𝟖 𝟐𝒚=𝟔𝒙 −𝟏𝟔
Procedures
3x −𝒚=𝟖
𝟐𝒚=𝟔𝒙−𝟏𝟔
Put each equation in Slope-
Intercept Form   
Identify the slope and y-
intercept
m = ____ b = ____
m = ____ b =____
3x - y = 8
- 3x x
-y = -3x + 8
y = 3x - 8
2y = 6x - 16
y = 3x - 8
-1( ) 𝟑 -8 𝟑 -8

16. Graph each equation in the graphing calculator and answer the following questions.
 Procedures
𝟑𝒙 −𝒚=𝟖
𝟐𝒚=𝟔𝒙 −𝟏𝟔
Identify the slope and y- intercept
m = ____ b = ____
m = ____ b =____ 𝟑 8 𝟑 8
What type of lines are there?
What type of solutions are there?
What do you notice about the equations when they are in slope intercept form AND do you think this has an effect on the solution?
Coinciding lines
Infinite many solutions
When the equations are in slope-intercept form, you notice the equations are the same. When the equations are the same, there are infinite many solutions.

17. Application of Systems of Equations

18. When you are given a word problem, remember to do the following:
Equation in Words
Variables
Equation with Numbers/Variables
Equation in Slope-Intercept Form 1) 2)
Rewrite the first equation algebraically using the variables chosen.
Rewrite the first equation in slope-intercept form and graph.
Determine the what the unknown variables are and what letter they will represent.
Rewrite what the first equation could be.
Rewrite the second equation algebraically using the variables chosen.
Rewrite the second equation in slope-intercept form and graph.
Rewrite what the second equation could be.
Finally, find the solution set using the calculator.

19. Equation with Numbers/Variables Equation in Slope-Intercept Form
The school is selling tickets to a band concert. On the first day of ticket sales, the school sold 3 adult tickets and 1 student tickets for a total of $28. The school took in $32 on the second day by selling 3 adult tickets and 2 student tickets. Find the price of a student ticket and an adult ticket.
Equation in Words
Variables
Equation with Numbers/Variables
Equation in Slope-Intercept Form 1) 2)
a = cost of adult tickets
(let a = x)
s = cost of student tickets
(let s = y)
3x + y = 28
- 3x x
y = -3x + 28
The school sold 3 adult tickets and 1 student ticket for $28
3x + y = 28
3x + 2y = 32
3x x
2y = -3x + 32
𝑦=− 3 2 𝑥+16
The school sold 3 adult tickets and 2 student ticks for $32
3x + 2y = 32

20. Equations in Slope-Intercept Form
The school is selling tickets to a band concert. On the first day of ticket sales, the school sold 3 adult tickets and 1 student tickets for a total of $28. The school took in $32 on the second day by selling 3 adult tickets and 2 student tickets. Find the price of a student ticket and an adult ticket.
Equations in Slope-Intercept Form
Graph of the System 1)
 y = -3x + 28 2)
 𝑦=− 3 2 𝑥+16
Solution set is (8, 4)

21. cost of student tickets
Graph each equation. Where is the solution set located? Remember what x and y equals when it relates the word problem: x = ______________ and y = _____________ What does the solution set mean?
The solution is located at (8, 4).
cost of adult tickets
cost of student tickets
Adult tickets cost $8 each and student tickets cost $4 each.

22. Equation with Numbers/Variables Equation in Slope-Intercept Form
You are getting ready to move and have asked some friends to help. For lunch, you buy the following sandwiches at the local deli for $30: six ham sandwiches and six turkey sandwiches. Later at night, everyone is hungry again and you buy four ham sandwiches and eight turkey sandwiches for $ What is the price of each sandwich?
Equation in Words
Variables
Equation with Numbers/Variables
Equation in Slope-Intercept Form 1) 2)
h = ham sandwiches
(let h = x)
t = tuna sandwiches
(let t = y)
6x + 6y = 30
- 6x x
6y = -6x + 30
y = -x + 5
6 ham and 6 turkey for $30
6x + 6y = 30
4x + 8y = 30.20
-4x x
8y = -4x
𝑦=− 1 2 𝑥+3.775
4 ham and 8 turkey for $30.60
4x + 8y = 30.20

23. Equations in Slope-Intercept Form
You are getting ready to move and have asked some friends to help. For lunch, you buy the following sandwiches at the local deli for $30: six ham sandwiches and six turkey sandwiches. Later at night, everyone is hungry again and you buy four ham sandwiches and eight turkey sandwiches for $ What is the price of each sandwich?
Equations in Slope-Intercept Form
Graph of the System 1)
 y = -x + 5 2)
 𝑦=− 1 2 𝑥+3.775
Solution set is (2, 3)

24. cost of turkey sandwiches
Graph each equation. Where is the solution set located? Remember what x and y equals when it relates the word problem: x = ______________ and y = _____________ What does the solution set mean?
The solution is located at (2, 3).
cost of ham sandwiches
cost of turkey sandwiches
Ham sandwiches cost $2 each and turkey sandwiches cost $3 each.

25. Equation with Numbers/Variables Equation in Slope-Intercept Form
The concession stand is selling hot dogs and hamburgers during a game. At halftime, they sold a total of 50 hot dogs and hamburgers and brought in $ How many of each item did they sell if hamburgers sold for $1.50 and hot dogs sold for $1.25?
Equation in Words
Variables
Equation with Numbers/Variables
Equation in Slope-Intercept Form 1) 2)
x + y = 50
- x x
y = -x + 50
50 hotdog and hamburgers were sold
h = hotdogs
(let h = x)
b = hamburgers
(let b = y)
x + y = 50
0.5x + y = 75.5
-0.5x x
y = -0.5x
With selling hotdogs for .50 and hamburgers for $1, $ was made.
0.5x + y = 75.50