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

Solving a system of equations by substitution Pick the easier equation. The goal is to get y= ; x= ; a= ; etc. Step 1: Solve an equation for one variable. Step 2: Substitute Put the equation solved in Step 1 into the other equation. Substitute the value of the variable into the equation. Step 3: Plug back in to find the other variable. Step 4: Check your solution. Substitute your ordered pair into BOTH equations.

1) Solve the system using substitution x + y = 5 y = 3 + x Step 1: Solve an equation for one variable. The second equation is already solved for y! Step 2: Substitute x + y = 5 x + (3 + x) = 5 2x + 3 = 5 2x = 2 x = 1

1) Solve the system using substitution x + y = 5 y = 3 + x x + y = 5 (1) + y = 5 y = 4 Step 3: Plug back in to find the other variable. (1, 4) (1) + (4) = 5 (4) = 3 + (1) Step 4: Check your solution. The solution is (1, 4). What do you think the answer would be if you graphed the two equations?

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

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

Elimination using Addition Consider the system x - 2y = 5 Lets add both equations to each other + 2x + 2y = 7 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 Lets substitute x = 4 into this equation. 2x + 2y = 7 4 - 2y = 5 Solve for y - 2y = 1 y = 1 2  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 Lets substitute x = 4 into this equation. 2x + 2y = 7 4 - 2y = 5 Solve for y - 2y = 1 1 2 y =  1 2 ANS: (4, ) 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 3x + y = 14 Substitute x = 3 into this equation 4x - 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.

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 = -3 12x + 20y = -8 When 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 -1 ( ) 6x + 11y = -5 6x + 9y = -3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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