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Lecture Comments (1)

### Graphing Systems of Equations

- A
*system of equations*is two equations in two variables. A*solution*consists of values for each unknown that satisfy both equations. - A system can have one, infinite, or no solutions.
- The system is called
*independent*if it has one solution. In this case, the graphs of the two equations intersect at one point. - It is called
*dependent*if it has infinitely many solutions. In this case, the graphs are the same line. - It is called
*inconsistent*if it has no solution. In this case, the graphs are parallel. - You can solve a system by graphing. Always check your solutions to be sure you have read the graph accurately.
- Use the method of graphing when you are willing to settle for an estimate for the solution.

### Graphing Systems of Equations

Solving the system by graphing

x + 2y = 35x + y = 5

x + 2y = 35x + y = 5

- Find two points for each equation by setting x = 0 then y = 0
- x + 2y = 3 → ( 0,[1/2] ),( 3,0 )
- 5x + y = 5 → ( 5.0, ),( 5,0 )

Graph to find intersection

(7, - 2)

(7, - 2)

Solving the system by graphing

y − x = 42y − x = 3

y − x = 42y − x = 3

- Find two points for each equation by setting x = 0 then y = 0
- y − x = 4 → ( 0,4 ),( − 4,0 )
- 2y − x = 3 → ( 0,[3/2] ),( − [3/2],0 )

Graph to find intersection

( - 5, - 1)

( - 5, - 1)

Solving the system by graphing

− x + 4y = 20y − 3x = 16

− x + 4y = 20y − 3x = 16

- Find two points for each equation by setting x = 0 then y = 0
- x + 4y = 20 → ( 0,5 ),( − 20,0 )
- y − 3x = 16 → ( 0, − [16/3] ),( 16,0 )

Graph to find intersection

( - 4,4)

( - 4,4)

Solving the system by graphing

− x + 6y = 12y + x = 9

− x + 6y = 12y + x = 9

- Find two points for each equation by setting x = 0 then y = 0
- − x + 6y = 12 → ( 0, − 12 ),( 2,0 )
- y + x = 9 → ( 9,0 ),( 0,9 )

Graph to find intersection

(6,3)

Solving the system by graphing

x + 2y = 16x + y = 8

x + 2y = 16x + y = 8

- Find two points for each equation by setting x = 0 then y = 0
- x + 2y = 16 → ( 0,8 ),( 16,0 )
- x − y = 8 → ( 8,0 ),( 0,8 )

Graph to find intersection

(0,8)

(0,8)

Solving the system by graphing

x + 2y = − 123y − x = 6

x + 2y = − 123y − x = 6

- Find two points for each equation by setting x = 0 then y = 0
- x + 2y = − 12 → ( − 12,0 ),( 0, − 6 )
- 3y − x = 6 → ( − 6,0 ),( 0,2 )

Graph to find intersection

( - 6,0)

Solving the system by graphing

x + y = 5y = − 3

x + y = 5y = − 3

- Find two points for each equation by setting x = 0 then y = 0
- x + y = 5 → ( 5,0 ),( 0,5 )
- y = − 3 → ( 0, − 3 ),( 0, − 3 )

Graph to find intersection

(8, - 3)

(8, - 3)

Solving the system by graphing

y − 3x = 2− 9x + 3y = 6

y − 3x = 2− 9x + 3y = 6

- Find two points for each equation by setting x = 0 then y = 0
- x + y = 5 → ( − [3/2],0 ),( 0,2 )
- y = − 3 → ( − [3/2],0 ),( 0,2 )

Graph to find intersection

Infinite solutions

Infinite solutions

Solving the system by graphing

2y − x = − 8y − x = − 3

2y − x = − 8y − x = − 3

- Find two points for each equation by setting x = 0 then y = 0
- 2y − x = − 8 → ( 0, − 4 ),( 8,0 )
- − y + x = 3 → ( 3,0 ),( 0, − 3 )

Graph to find intersection

( - 2, - 5)

Solving the system by graphing

y + 2x = 5x + [y/2] = 2

y + 2x = 5x + [y/2] = 2

- Find two points for each equation by setting x = 0 then y = 0
- y + 2x = 5 → ( 0,5 ),( [5/2],0 )
- x + [y/2] = 2 → ( 2,0 ),( 0,4 )

Graph to find intersection

No solutions

No solutions

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Graphing Systems of Equations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Systems of Equations 0:10
- Definition
- Example
- Solution
- Solving by Graphing 1:23
- Points of Intersection
- Example
- Number of Solutions 3:09
- Independent
- Dependent
- Inconsistent
- Example 1: Solve by Graphing 5:45
- Example 2: Solve by Graphing 9:50
- Example 3: Solve by Graphing 14:17
- Example 4: Solve by Graphing 18:03