Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or...
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![Page 1: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/1.jpg)
Solving Systems of Linear Equations by Graphing
![Page 2: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/2.jpg)
Definitions
• A system of linear equations is two or more linear equations.
• Ex:
Solution of a system of linear equations in 2 variables is an ordered pair of numbers that is a solution of both equations in the system.
Example: (0,-4)
![Page 3: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/3.jpg)
How can we find the solution of a system of linear equations?
• Graphing-• Graph each equation
and see where the lines intersect!
• Graph the system:
• Y = x + 1 and y = 2x - 1
• When we graph we graph on the same coordinate system!
![Page 4: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/4.jpg)
![Page 5: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/5.jpg)
• How do we determine if our graph is correct?
• Substitute the ordered pair on the graph to check and make sure it is a solution
• Y = x + 1• Y = 2x -1
![Page 6: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/6.jpg)
• Example: 3x + 4y = 12
9x + 12y = 36
Solution for the same line :
Infinite amount of solutions!
![Page 7: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/7.jpg)
• Example: 3x – y = 66x = 2y
Lines that are parallel do not have a solution:
Answer: No solution!
![Page 8: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/8.jpg)
• How can we determine whether or not we have a system with infinite amount of solutions or no solution?
• Using our slope and y intercepts!
• To help you find the solution, before graphing write each equation in slope intercept form!
![Page 9: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/9.jpg)
• If the slopes are the same and the y intercepts are the same, then you will have an infinite amount of solutions!
• IF the slopes are the same and the y intercepts are different, then you will have parallel lines!
• If the slopes are different, then you will have one solution, an ordered pair!
![Page 10: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/10.jpg)
Let’s go back and check our examples!
3x + 4y = 12-3x -3x
4y = -3x + 124 4 4
y = -3x + 3 4
• 9x + 12y = 36-9x -9x
12y = -9x + 3612 12 12
y = -3x + 3 4
![Page 11: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/11.jpg)
3x – y = 6-3x -3x
-y = -3x + 6 -1 -1
Y = 3x - 6
• 6x = 2y 2 2
Y = 3x or y = 3x + 0
![Page 12: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/12.jpg)
Different Types of Systems
• Consistent Systems: has at least one solution
• Inconsistent Systems: have no solution
![Page 13: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/13.jpg)
Different Types of Equations
• Independent equations:Different types of linear
equations (not the same line)
• Dependent Equations: the exact same graph
• P. 247
![Page 14: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/14.jpg)
Solving Systems of Linear Equations
![Page 15: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/15.jpg)
Definitions
• A system of linear equations is two or more linear equations.
• Solution of a system of linear equations in 2 variables is an ordered pair of numbers that is a solution of both equations in the system.
![Page 16: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/16.jpg)
How can we determine what the solution is?
• Guess/Check• Graphing• Substitution• Elimination
![Page 17: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/17.jpg)
Graphing
![Page 18: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/18.jpg)
Guess and Check
• Subsitute all the choices into BOTH equations!!!!
• If the ordered pair is true for both equations then it is a system of the set of linear equations!
• 2x – y = 8• X + 3y = 4
a). (3, -2)b). (-4, 0)c). (0, 4)d). (4,0)
![Page 19: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/19.jpg)
Example:
-3x + y = -10X – y = 6
a). (-2, 4)b). (2, 4)c). (2, -4)
![Page 20: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/20.jpg)
3x + 4y = 129x + 12y = 36
a). (0,3)b). (-4,0)c). (-4, 6)
![Page 21: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/21.jpg)
• Systems of linear equations can have MORE THAN ONE SOLUTION!
• These type of systems have an Infinite amount of solutions!
• Why?
![Page 22: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/22.jpg)
• Y = x – 3• 2y = 2x – 6
Let’s try graphing!*Write the equation in
y = mx + 6What is the slope?
What is the y intercept?
• It is the exact same equation!!!!!!
• Therefore it is the exact same line and it will intersect at every single point!
![Page 23: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/23.jpg)
• 2x – 3y = 6• -4x + 6y = 5
• Again, let’s write our equation in y=mx + b
• What is the slope of each equation and the y-intercept?
• Try graphing!
• Equations that have the same slope and different y-intercepts are parallel!
• They have NO SOLUTION!!!!
![Page 24: Solving Systems of Linear Equations by Graphing. Definitions A system of linear equations is two or more linear equations. Ex: Solution of a system of.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649e185503460f94b0475e/html5/thumbnails/24.jpg)
Summary!
• A system of linear equations can have three different solutions– NO solution : the lines are parallel to each (they
have the same slope and different y-intercepts)– Infinite amount of solutions: The lines are the
same (they have the same slope and same y-intercept)
– One solution: Our answer is an ordered pair!