Solving Systems of Equations The Elimination Method Three

Three Ways to solve a system • Graphing Method • Substitution Method • Elimination

Review Quickly – Solve System using Substitution 1. ) x + 4 y =

Elimination using Addition Consider the system x - 2 y = 5 2 x

Elimination using Addition Consider the system x - 2 y = 5 + 2

Elimination using Addition Consider the system 3 x + y = 14 4 x

Elimination using Addition Consider the system 3 x + y = 14 + 4

Elimination using Addition Consider the system 3 x + y = 14 Substitute x

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

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

Elimination using Multiplication Consider the system -1 (6 x + 11 y = -5

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

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

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

Elimination using Multiplication Consider the system + -3 x + -6 y = -18

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

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

More complex Problems Consider the system + 6 x + 8 y = -50

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

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Solving Systems of Equations The Elimination Method

Three Ways to solve a system • Graphing Method • Substitution Method • Elimination Method

Review Quickly – Solve System using Substitution 1. ) x + 4 y = 8 2 x + 3 y = 1 Step 1 x + 4 y = 8 -4 y x = -4 y +8 2. ) -3 x + 2 y = -11 5 x – y = 23 Step 2 Step 1 2(-4 y + 8) + 3 y = 1 -8 y + 16 + 3 y = 1 -5 y + 16 = 1 -16 -5 y = -15 y=3 Answer: (-4, 3) Answer: (5, 2) Step 2

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

Elimination using Addition Consider the system x - 2 y = 5 + 2 x + 2 y = 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 - 2 y = 5 + 2 x + 2 y = 7 = 12 3 x x=4 Lets add both equations to each other 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 - 2 y = 5 2 x + 2 y = 7 4 - 2 y = 5 - 2 y = 1 y= 1 2 Lets substitute x = 4 into this equation. (either equation works) Solve for y 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 - 2 y = 5 2 x + 2 y = 7 4 - 2 y = 5 - 2 y = 1 y= 1 2 Lets substitute x = 4 into this equation. Solve for y 1 ANS: (4, 2 ) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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

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

Elimination using Addition Consider the system 3 x + y = 14 Substitute x = 3 into this equation 4 x - y = 7 3(3) + y = 14 9 + y = 14 y=5 ANS: (3, 5 ) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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

Elimination using Multiplication Consider the system + 6 x + 11 y = -5 6 x + 9 y = -3 12 x + 20 y = -8 When we add equations together, nothing cancels out

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

Elimination using Multiplication Consider the system -1 (6 x + 11 y = -5 ) 6 x + 9 y = -3

Elimination using Multiplication Consider the system + - 6 x - 11 y = 5 6 x + 9 y = -3 -2 y = 2 y = -1 ANS: (x, -1 )

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

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

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

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

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

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

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

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

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

More complex Problems Consider the system + 6 x + 8 y = -50 -6 x + 9 y = -18 17 y = -68 y = -4 ANS: (x, -4)

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

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

Examples 1. x + 2 y = 5 2. x + 2 y = 4 2 x + 6 y = 12 x - 4 y = 16 ANS: (3, 1) ANS: (8, -2)

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

Examples… 2. 1. 4 x + y = 9 2 x + 3 y = 1 3 x + 2 y = 8 5 x + 7 y = 3 ANS: (2, 1) ANS: (2, -1)