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Warm-Up
CAHSEE/CCSS: 7 AF 1.1/ A–REI.17 Review/CCSS: ALG 6.0/ 8.EE.
Which system of equations
represents the statements below?
Graph the following linear
equation in two ways.
– 2 x + 3 y = 8
Current/CCSS: ALG 6.0/ 8.F.4 Other/CCSS: ALG 5.0/ 8.EE.7a
What equation best represents
the line shown in the graph below?
Write the equation in two
different ways.
Solve the following equation in
three different ways.
3 ( x + 5 ) = 2 x + 35
y
x
The sum of two numbers is ten.
One number is five times the other.
I. Definition
- Define system of equations as “a set of 2 or more equations with the same variables.”
II. Introduction to New Material
Ex. 1
- Graph x + y = – 4 by finding the x and y intercepts
- Graph – 2 x + y = 2 on the same axes
- Ask students,
o “How many times do the lines intersect?” [One!]
o “The number of times that the lines intersect is the number of solutions this system has.
How many solutions does this system have?” [One!]
o “Where is this solution (The point where the lines intersect)?” (–2, – 2 )
- In the solutions column (in student notes) have students write:
o One solution
o Lines only intersect one time
o “What do we know about the slopes of these two lines? Are they the same or different?”
[Different]
o “What about the y - intercepts?” [Different]
- Add this information to the “solutions” column
o One solution
o Lines only intersect one time
o Slopes are different
o y- intercepts are different
Ex. 2 and 3 – Follow the above process.
III. Think, Pair, Share
- Give students time to graph the system of equations and answer the questions about the graph
independently.
- Afterwards, instruct students to compare their work with their partner’s.
o “Turn to your partner and check if your graphs looks the same. If not, see who made a
mistake. Tell your partner how you answered the 3 questions and explain how you got those
answers.”
- Have pairs of students come up to the document camera to share graphs and answers.
IV. “Your Turns” – Independent Practice Match
- Give each student (or pair of students) an envelope of the 18 pre-cut matching pieces.
- Instructions: “Match system number with graph letter and the system’s correct solution. Provide
justification as to why the three cards belong together.”
o Example: “System 3 (
st
column) might match up with graph F (
nd
column) which could
have no solution (
rd
column). Then provide the explanation.”
III. Graph the following system and answer the questions about its solution.
System Graph Solution
3 x + 2 y = – 6
- 3 x + 2 y = 6
IV. Matching
System of Equations
(Write the number)
Graph
(Write the letter)
Solution Justification
!
How many times do these
lines intersect?
Where do they intersect?
What is the solution to the
system?
Solving a System by Graphing
I. System of Equations: a set of 2 or more equations with the same variables.
II. A system of equations can have 3 types of solutions.
System Graph Solution
Ex 1.
x + y = – 4
- 2 x + y = 2
x + y = – 4 0 + y = – 4 y = – 4 ( 0 , – 4 ) x + y = – 4 x + 0 = – 4 x = – 4 (– 4 , 0 )
- 2 x + y = **2
- 2 ( 0 )** + y = 2 y = **2 ( 0 , 2 )
- 2** x + y = **2
- 2** x + 0 = 2 - 2 x = 2 x = – 1 (– 1 , 0 )
Ex 2.
- x + 2 y = 2
3 x – 6 y = 12
- x + 2 y = 2 0 + y = 2 y = 1 ( 0 , 1 ) - x + 2 y = 2 - x + 2 ( 0 ) = 2 - x = 2 x = – 2 (– 2 , 0 ) 3 x – 6 y = 12 3 ( 0 ) – 6 y = **12
- 6** y = 12 y = – 2 ( 0 , – 2 ) 3 x – 6 y = 12 3 x – 6 ( 0 ) = 12 3 x = 12 x = 4 ( 4 , 0 )
Ex 3.
- 8 x + 2 y = 8
4 x – y = – 4
- 8 x + 2 y = **8
- 8 ( 0 )** + 2 y = 8 2 y = 8 y = **4 ( 0 , 4 )
- 8** x + 2 y = **8
- 8** x + 2 ( 0 ) = **8
- 8** x = 8 x = – 1 (– 1 , 0 ) 4 x – y = – 4 4 ( 0 ) – y = – 4 - y = – 4 y = 4 ( 0 , 4 ) 4 x – y = – 4 4 x – 0 = – 4 4 x = – 4 x = – 1 (– 1 , 0 )
One Solution
once (–2, – 2)
- Slopes are different - y - intercepts are
different
No Solution
(parallel lines)
- Slopes are the same - y - intercepts are
different
Infinite Solutions
every point
(coinciding lines)
- Slopes are the same - y - intercepts are
the same
1 3 x − y = 2 − 6 x + 2 y = 6 "
$ C No Solution 2 2 x − y = − 1 3 x + y = 6 "
$ D One Solution (1, 3) 3 x + y = 5 − 2 x − 2 y = − 10 "
$ A Infinite Solutions
4 3 x + 2 y = 6 6 x + 4 y = 6 ! "
F No Solution 5 3 x + 2 y = 4 − x + y = − 3 "
$ B One Solution (2, - 1 ) 6 3 x + 2 y = 6 − 6 x − 4 y = − 12 ⎧ ⎨ ⎩ E Infinite Solutions
