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

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.

### System of Linear Equations

### A Final way to Solve Systems:

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

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 SubstitutionHere’s the problem: Equation one -x+y=1 Equation two 2x+y=-2

Example: The Substitution MethodFirst, 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

- 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 = 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) = 46

12x+20y = 24 --------

3 ; 3(-4x+2y) = 35

-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 ( )

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 two 2x+y=-2

Step 1 Download Application Quick graph from programe App Store.

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

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 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

- 4. Use linear combinations to solve this system form y=-2-2x
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 form y=-2-2xExercises

- 1. (2,-1)
- 2. (-1,3)
- 3. (1,2)
- 4. (3,1)
- 5. (-0.5, 0.5)

The End form y=-2-2x

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