Good morning everybody we will take you on a fun learning today
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Good morning everybody . we will take you on a fun learning today. Mrs. Panchaporn Kantayasakun. New generation S.W.K. Mrs. Khrongsri Nampoon. By :. Mrs. Jarin Promsri. System of Linear Equations.

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Good morning everybody we will take you on a fun learning today

Good morning everybody.we will take you on a fun learning today.


New generation s w k

Mrs. PanchapornKantayasakun

New generation S.W.K

Mrs. KhrongsriNampoon

By:

Mrs. JarinPromsri


System of linear equations

System of Linear Equations

How to: solve by graphing, substitution, linear combinations, and special types of linear systems


What is a linear system anyways

What is a Linear System, Anyways?

  • A linear system includes two, or more, equations, and each includes two or more variables.

  • When two equations are used to model a problem, it is called a linearsystem.


Before you begin important terms to know

Before You Begin…Important Terms to know

  • Linear system: two equations that form one equation

  • Solution: the answer to a system of linear equation; must satisfy both equations***: a solution is written as an ordered pair: (x,y)

  • Leading Coefficient: any given number that is before any given variable (for example, the leading coefficient in 3x is 3.)

  • Isolate: to get alone


Good morning everybody we will take you on a fun learning today

Ways to Solve Linear Systems

By: Substitution


Solving linear systems by substitution

Basic steps:

1. Solve one equation for one of its variables

2. Substitute that expression into the other equation and solve for the other variable

3. Substitute that value into first equation; solve

4. Check the solution

Solving Linear Systems by Substitution


Example the substitution method

Here’s the problem:Equation one-x+y=1Equation two2x+y=-2

Example: The Substitution Method


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First, solve equation one for y

y = x+1

Next, substitute the above expression in for “y” in equation two, and solve for x

Here’s how:

Equation two

2x+y=-2

2x+ (x+1)=-2

3x+1=-2

3x=-3

x=-1


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Congratulations! You now know x has a value of –1…but you still need to find “y”.

To do so…

First, write down equation one

y=x+1

y= (-1)+1

y=0

So, now what?

You’re done; simply write out the solution as (-1,0)

***Did you remember?

To write a solution, once you’ve found x and y, you must put x first and then y: (x,y)


Solving linear systems by linear combinations

Solving Linear Systems by Linear Combinations


Solving systems by means of linear combinations

Solving Systems by means of Linear Combinations

  • Basic steps:1. Arrange the equations with like terms in columns

    2. After looking at the coefficients of x and y, you need to multiply one or both equations by a number that will give you new coefficients for x or y that are opposites.

    3. Add the equations and solve for the unknown variable

    4. Substitute the value gotten in step 3 into either of the original equations; solve for other variable

    5. Check the solution in both original equations


Example solving systems by linear combinations

Example: Solving Systems by Linear Combinations

  • Solve this linear equation:Equation One: 3x+5y = 6Equation Two: -4x+2y =5


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Solve the linear system

Equation 1: 3x+5y=6

Equation 2: -4x+2y=5

3x+5y= 6 --------

-4x+2y= 5 --------

4;4(3x+5y)= 46

12x+20y= 24 --------

3 ;3(-4x+2y) = 35

-12x +6y= 15 --------

+ ; 12x+ 20y -12x + 6y = 24 + 15

26y = 39


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Equation 2: -4x+2y=5

Substitute the value you just found for

-4x+2( ) = 5

-4x+3 = 5

-4x = 2

x =

The solution to the example system is ( )


A final way to solve systems

A Final way to Solve Systems:

Graph and Check


Types of solutions of systems of equations

Types of Solutions of Systems of Equations

  • One solution – the lines cross at one point

  • No solution – the lines do not cross

  • Infinitely many solutions – the lines coincide


An example of the quick graph on and check method

An Example of the Quick graph on and Check Method

  • Here’s the problem:

    Equation one-x+y=1

    Equation two2x+y=-2


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Step 1 Download Application Quick graph from programe App Store.


Step 2 open app quick graph on iphone or samsung etc

Step 2 open App Quick graph on Iphone or samsung etc.


Step 3 click the plus sign

Step 3 Click the plus sign.


Step 4 type the equation in the form y 1 x and click done

Step 4 Type the equation in the form y=1+x and click Done .


Step 5 will have a graph

Step 5 will have a graph


Step 6 click the plus sign and type the equation in the form y 2 2x and click done

Step 6 Click the plus sign. And Type the equation in the form y=-2-2xand click Done


Step 7 will have a graph for equation y 2 2x

Step 7 will have a graph for equation y=-2-2x

Answer to the equation is the graph intersect (-1,0)


Fun fun exercises

Fun, Fun: Exercises

  • 1. Solve the following Linear System

    Equation one: 3x-4y=10

    Equation two: 5x+7y=3

  • 2. Solve the following Linear system

    Equation one: x-6y=-19

    Equation two: 3x-2y=-9

  • 3. Solve the following Linear system

    Equation one: x+3y=7

    Equation two: 4x-7y=-10


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  • 4. Use linear combinations to solve this system

    Equation one: x+2y=5

    Equation two: 3x-2y=7

  • 5. Use linear combinations to solve this system

    Equation one: 3x-5y=-4

    Equation two: -9x+7y=8


Answers to the exercises

Answers to the Exercises

  • 1. (2,-1)

  • 2. (-1,3)

  • 3. (1,2)

  • 4. (3,1)

  • 5. (-0.5, 0.5)


Check out the answers from of the quick graph

Check out the answers from of the Quick graph

1.

3x-4y=10

5x+7y=3


Check out the answers from of the quick graph1

Check out the answers from of the Quick graph

2.

x-6y=-19

3x-2y=-9


Check out the answers from of the quick graph2

Check out the answers from of the Quick graph

3.

x+3y=7

4x-7y=-10


Check out the answers from of the quick graph3

Check out the answers from of the Quick graph

4.

x+2y=5

3x-2y=7


Check out the answers from of the quick graph4

Check out the answers from of the Quick graph

5.

3x-5y=-4

-9x+7y=8


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The End


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