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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



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=1 Equation two 2x+y=-2

Example: The Substitution Method


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


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 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 = 6 Equation Two: -4x+2y =5


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


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 ( )



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 two 2x+y=-2



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






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

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


Fun fun exercises
Fun, Fun: form y=-2-2xExercises

  • 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



Answers to the exercises
Answers to the form y=-2-2xExercises

  • 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 form y=-2-2x

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 form y=-2-2x

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 form y=-2-2x

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 form y=-2-2x

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 form y=-2-2x

5.

3x-5y=-4

-9x+7y=8


The End form y=-2-2x


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