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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.

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

Mrs. PanchapornKantayasakun

## New generation S.W.K

Mrs. KhrongsriNampoon

By:

Mrs. JarinPromsri

## System of Linear Equations

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

### 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

• 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

Ways to Solve 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

Here’s the problem:Equation one-x+y=1Equation two2x+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 Linear Systems by 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

• Solve this linear equation:Equation One: 3x+5y = 6Equation 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 ( )

## A Final way to Solve Systems:

Graph and Check

### 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

• Here’s the problem:

Equation one-x+y=1

Equation two2x+y=-2

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

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

### 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

• 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

• 1. (2,-1)

• 2. (-1,3)

• 3. (1,2)

• 4. (3,1)

• 5. (-0.5, 0.5)

1.

3x-4y=10

5x+7y=3

2.

x-6y=-19

3x-2y=-9

3.

x+3y=7

4x-7y=-10

4.

x+2y=5

3x-2y=7

5.

3x-5y=-4

-9x+7y=8

The End