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# Good morning everybody . we will take you on a fun learning today. - PowerPoint PPT Presentation

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.

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

• 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.

• 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

By: Substitution

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=1 Equation two 2x+y=-2

Example: The Substitution Method

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

• 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

• 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 ( )

### A Final way to Solve Systems:

Graph and Check

• One solution – the lines cross at one point

• No solution – the lines do not cross

• Infinitely many solutions – the lines coincide

• 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 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: 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

• 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 form y=-2-2x

1.

3x-4y=10

5x+7y=3

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 graph form y=-2-2x

3.

x+3y=7

4x-7y=-10

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 graph form y=-2-2x

5.

3x-5y=-4

-9x+7y=8

The End form y=-2-2x