Systems of linear equations
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Systems of Linear Equations!. By graphing. Definition. A system of linear equations, aka linear system, consists of two or more linear equations with the same variables. x + 2y = 7 3x – 2y = 5. The solution.

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Presentation Transcript

Definition
Definition

  • A system of linear equations, aka linear system, consists of two or more linear equations with the same variables.

    • x + 2y = 7

    • 3x – 2y = 5


The solution
The solution

  • The solution of a system of linear equations is the ordered pair that satisfies each equation in the system.

  • One way to find the solution is by graphing.

    • The intersection of the graphs is the solution.


Example
Example

X + 2y = 7

3x – 2y = 5

  • Step 1: graph both equations

  • Step 2: estimate coordinates of the intersection

  • Step 3: check algebraically by subsitution


Types of systems
Types of systems

Consistent Independent System – has exactly one solution

*other types to be discussed later


More examples
More examples

-5x + y = 0

5x + y = 10

-x + 2y = 3

2x + y = 4


Multi step problem
Multi-step problem

x + y = 25

15x + 30y = 450

A business rents in line skates ad bicycles. During one day the businesses has a total of 25 rentals and collects $450 for the rentals. Find the total number of pairs of skates rented and the number of bicycles rented.

Skates - $15 per day

Bikes - $30 per day




Steps
Steps made $420.

3x – y = -2

X + 2y = 11

3x + 2 = y

X + 2(3x + 2) = 11

Step 2: substitute the expression in the other equation for the variable and solve

X + 6x + 4 = 11

7x = 7

X = 1

Step 3: substitute the solution back into the equation from step 1 and solve

3(1) + 2 = y

5 = y

Solution: (1,5)

Step 1: Solve one of the equations for a variable


More examples1
More examples made $420.

X – 2y = -6

4x + 6y = 4

Y = 2x + 5

3x + y = 10

3x + y = -7

-2x + 4y = 0


Multi step problem1
Multi-step problem made $420.

X + y = 26

15x + 7.5y = 360

A group of friends takes a day-long tubing trip down a river. The company that offers the tubing trip charges $15 to rent a tube for a person to use and $7.50 to rent a tube to carry the food and water in a cooler. The friends spend $360 to rent a total of 26 tubes. How many of each type of tube do they rent?


Elimination

Elimination made $420.

7.3


Elimination method
Elimination Method made $420.

2x + 3y = 11

-2x + 5y = 13

8y = 24

(1,3)

Step 2: Solve the resulting equation for the other variable.

8y = 24

Y = 3

Step 3: Substitute into either original equation to find the value of the other variable.

2x + 3(3) = 11

2x + 9 = 11

2x = 2

X = 1

Step 1: Add the equations to eliminate one variable.


A little twist
A little twist made $420.

Step P: Make Opposite

Step 1: Add

Step 2: Solve

Step 3: Substitute/Solve

4x + 3y = 2

5x + 3y = -2

4x + 3y = 2

-5x – 3y = 2

-1( )

-x = 4

(-4, 6)

X = -4

4(-4) + 3y = 2

-16 + 3y = 2

3y = 18

Y = 6


Arranging like terms
Arranging like terms made $420.

If two linear systems are not in the same form you must rearrange one!

8x – 4y = -4

4y = 3x + 14


Examples
Examples made $420.

You try:

4x – 3y = 5

-2x + 3y = -7

-5x – 6y = 8

5x + 2y = 4

3x + 4y = -6

2y = 3x + 6

7x – 2y = 5

7x – 3y = 4

2x + 5y = 12

5y = 4x + 6


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