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 linear equations in 2 variables is an ordered pair of numbers that is a solution of both equations in the system. Example: (0,-4)

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!

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

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

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

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!

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!

Let’s go back and check our examples! 3x + 4y = 12 -3x -3x 4y = -3x + 12 4 4 4 y = -3x + 3 4 9x + 12y = 36 -9x -9x 12y = -9x + 36 12 12 12 y = -3x + 3 4

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

Different Types of Systems Consistent Systems: has at least one solution Inconsistent Systems: have no solution

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

Solving Systems of Linear Equations

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.

How can we determine what the solution is? Guess/Check Graphing Substitution Elimination

Graphing

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)

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

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

Systems of linear equations can have MORE THAN ONE SOLUTION! These type of systems have an Infinite amount of solutions! Why?

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!

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!!!!

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!