Solving Systems of Equations Elimination and Substitution Elimination

Solving Systems of Equations Elimination and Substitution

Elimination Using Addition Consider the system 2 x - 2 y = -8 2 x + 2 y = 4 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 2 x - 2 y = -8 + 2 x + 2 y = 4 Lets add both equations to each other NOTE: We use the Elimination Method, if we can immediately cancel two like terms.

Elimination Using Addition Consider the system 2 x - 2 y =-8 + 2 x + 2 y = 4 = -4 4 x x = -1 Lets add both equations to each other ANS: (-1, y) NOTE: We use the Elimination Method, if we can immediately cancel two like terms.

Elimination Using Addition Consider the system 2 x - 2 y = -8 2 x + 2 y = 4 2(-1) - 2 y = -8 -2 y = -6 y=3 Lets substitute x = -1 into this equation. Solve for y ANS: (-1, y) NOTE: We use the Elimination Method, if we can immediately cancel two like terms.

Elimination Using Addition Consider the system 2 x - 2 y = -8 2 x + 2 y = 4 2(-1) - 2 y = -8 -2 y = 6 y =3 Lets substitute x = -1 into this equation. Solve for y ANS: (-1, 3 ) NOTE: We use the Elimination Method, if we can immediately cancel two like terms.

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

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

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 the equations together, nothing cancels.

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 =8 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)

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)

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 = -6 x = -3 Substitute y = -4 ANS: ( -3 , -4)

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)

Substitution Method Consider the system: Solve this equation for x. X = -2 y - 1 Substitute this into the other equation.

Substitution Method Solve this equation for x. 4(-2 y – 1) – 3 y = -15 -8 y -4 -3 y = -15 -11 y – 4 = -15 -11 y = -11 y=1 Substitute (-2 y - 1) into the other equation. ANS: ( x, 1)

Substitution Method Substitute y = 1 into this equation. 4 x – 3(1) = -15 4 x – 3 = -15 4 x = -12 X = -3 ANS (-3, 1)