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Graphing and Solving Systems of Linear Equations: Exercises and Solutions, Lecture notes of Linear Algebra

A step-by-step guide for graphing and solving systems of linear equations. It includes examples, think-pair-share activities, and independent practice exercises. Students are asked to identify the number of solutions, the intersection point, and the slopes and y-intercepts of the lines.

What you will learn

  • How do you identify the intersection point of two lines?
  • What is the process for graphing a system of linear equations?
  • How do you find the number of solutions for a system of linear equations?

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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Page 1 of 8 MCC@WCCUSD 10/29/12
Warm-Up
CAHSEE/CCSS: 7 AF 1.1/ A–REI.17
Review/CCSS: ALG 6.0/ 8.EE.5
Which system of equations
represents the statements below?
Graph the following linear
equation in two ways.
–2x + 3y = 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.
3x+5
( )
=2x+35
y
x
The sum of two numbers is ten.
One number is five times the other.
pf3
pf4
pf5
pf8

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Download Graphing and Solving Systems of Linear Equations: Exercises and Solutions and more Lecture notes Linear Algebra in PDF only on Docsity!

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

  • Ask students,

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

  • Lines only intersect

once (–2, – 2)

  • Slopes are different - y - intercepts are

different

No Solution

  • Lines never intersect

(parallel lines)

  • Slopes are the same - y - intercepts are

different

Infinite Solutions

  • Lines intersect at

every point

(coinciding lines)

  • Slopes are the same - y - intercepts are

the same

1 3 xy = 2 − 6 x + 2 y = 6 "

$ C No Solution 2 2 xy = − 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