Solving Systems of Equations The Elimination Method.

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Presentation transcript:

Solving Systems of Equations The Elimination Method

Objectives Learn the procedure of the Elimination Method using addition Learn the procedure of the Elimination Method using multiplication Solving systems of equations using the Elimination Method

Elimination using Addition Consider the system x - 2y = 5 2x + 2y = 7 REMEMBER: We are trying to find the Point of Intersection. (x, y) Lets add both equations to each other

Elimination using Addition Consider the system x - 2y = 5 2x + 2y = 7 Lets add both equations to each other + NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition Consider the system x - 2y = 5 2x + 2y = 7 Lets add both equations to each other + 3x = 12 x = 4  ANS: (4, y) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition Consider the system x - 2y = 5 2x + 2y = 7 ANS: (4, y) Lets substitute x = 4 into this equation y = 5Solve for y - 2y = 1 y = 1 2  NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition Consider the system x - 2y = 5 2x + 2y = 7 ANS: (4, ) Lets substitute x = 4 into this equation y = 5Solve for y - 2y = 1 y = 1 2  1 2 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition Consider the system 3x + y = 14 4x - y = 7 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Elimination using Addition Consider the system 3x + y = 14 4x - y = 7 7x= 21 x = 3  ANS: (3, y) +

Elimination using Addition Consider the system ANS: (3, ) 3x + y = 14 4x - y = 7 Substitute x = 3 into this equation 3(3) + y = y = 14 y = 5 5  NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

Examples… 1.2. ANS: (4, -3)ANS: (-1, 2)

Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = -3

Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = x + 20y = -8When we add equations together, nothing cancels out

Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = -3

Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = ( )

Elimination using Multiplication Consider the system - 6x - 11y = 5 6x + 9y = y = 2 y = -1 ANS: (x, ) 

Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = -3 ANS: (x, ) y = -1 Lets substitute y = -1 into this equation 6x + 9(-1) = -3 6x + -9 = x = 6 x = 1 

Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = -3 ANS: (, ) y = -1 Lets substitute y = -1 into this equation 6x + 9(-1) = -3 6x + -9 = x = 6 x = 1 1 

Elimination using Multiplication Consider the system x + 2y = 6 3x + 3y = -6 Multiply by -3 to eliminate the x term

Elimination using Multiplication Consider the system x + 2y = 6 3x + 3y = ( )

Elimination using Multiplication Consider the system -3x + -6y = -18 3x + 3y = y = -24 y = 8 ANS: (x, 8) 

Elimination using Multiplication Consider the system x + 2y = 6 3x + 3y = -6 ANS: (x, 8) Substitute y =14 into equation y =8 x + 2(8) = 6 x + 16 = 6 x = -10 

Elimination using Multiplication Consider the system x + 2y = 6 3x + 3y = -6 ANS: (, 8) Substitute y =14 into equation y =8 x + 2(8) = 6 x + 16 = 6 x = 

Examples 1. x + 2y = 5 2x + 6y = ANS: (3, 1) x + 2y = 4 x - 4y = 16 ANS: (8, -2)

More complex Problems Consider the system 3x + 4y = -25 2x - 3y = 6 Multiply by 2 Multiply by -3

More complex Problems Consider the system 3x + 4y = -25 2x - 3y = 6 2( ) -3( )

More complex Problems Consider the system 6x + 8y = x + 9y = y = -68 y = -4 ANS: (x, -4) 

More complex Problems Consider the system 3x + 4y = -25 2x - 3y = 6 ANS: (x, -4) Substitute y = -4 2x - 3(-4) = 6 2x = 6 2x + 12 = 6 2x = -6 x = -3 

More complex Problems Consider the system 3x + 4y = -25 2x - 3y = 6 ANS: (, -4) Substitute y = -4 2x - 3(-4) = 6 2x = 6 2x + 12 = 6 2x = -6 x = -3  -3

Examples… x + y = 9 3x + 2y = 8 2x + 3y = 1 5x + 7y = 3 ANS: (2, 1)ANS: (2, -1)