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**Solving Systems of Linear Equations by Graphing**

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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)

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**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!

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**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

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Example: 3x + 4y = 12 9x + 12y = 36 Solution for the same line : Infinite amount of solutions!

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Example: 3x – y = 6 6x = 2y Lines that are parallel do not have a solution: Answer: No solution!

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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!

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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!

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**Let’s go back and check our examples!**

3x + 4y = 12 -3x x 4y = -3x + 12 y = -3x + 3 4 9x + 12y = 36 -9x -9x 12y = -9x + 36 y = -3x + 3 4

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3x – y = 6 -3x x -y = -3x + 6 -1 -1 Y = 3x - 6 6x = 2y Y = 3x or y = 3x + 0

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**Different Types of Systems**

Consistent Systems: has at least one solution Inconsistent Systems: have no solution

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**Different Types of Equations**

Independent equations: Different types of linear equations (not the same line) Dependent Equations: the exact same graph P. 247

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**Solving Systems of Linear Equations**

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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.

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**How can we determine what the solution is?**

Guess/Check Graphing Substitution Elimination

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Graphing

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**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)

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**Example: -3x + y = -10 X – y = 6 a). (-2, 4) b). (2, 4) c). (2, -4) **

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**3x + 4y = 12 9x + 12y = 36 a). (0,3) b). (-4,0) c). (-4, 6) **

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**Systems of linear equations can have MORE THAN ONE SOLUTION!**

These type of systems have an Infinite amount of solutions! Why?

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Y = x – 3 2y = 2x – 6 Let’s try graphing! *Write the equation in y = mx + 6 What 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!

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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!!!!

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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!

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