3.2 solving systems algebraically 2. solving systems by elimination

3.2 Solving Systems Algebraically

2. Solving Systems by Elimination

2) Solving Systems by Elimination

• By adding and subtracting linear systems, you can “eliminate” a variable and solve for an unknown

2) Solving Systems by Elimination

Example 1:

Solve the system by elimination.

4x + 3y = 4

2x – y = 7{

2) Solving Systems by Elimination

Example 1:

Solve the system by elimination.

4x + 3y = 4

2x – y = 7{Elimination – add or subtract the equations of a linear system until you “eliminate” a variable

2) Solving Systems by Elimination

Example 1:

Solve the system by elimination.

4x + 3y = 4 1

2x – y = 7 2

Step 1: Number the equations.

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2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4 1

2x – y = 7 2

Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations.

{

2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4 1

2x – y = 7 2 multiply by 2

Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations.

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2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4 1

2x – y = 7 2 multiply by 2

4x – 2y = 14

Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations.

{2

2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4 1

2x – y = 7 2 multiply by 2

4x – 2y = 14

Step 3: Equation subtract

{2

21

2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4

4x – 2y = 14

Step 3: Equation subtract 21

1

2-

Use subtraction to eliminate x

2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4

4x – 2y = 14

5y = -10

y = -2

Step 3: Equation subtract 21

1

2-

2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4

4x – 2y = 14

5y = -10

y = -2

Step 4: Use substitution to solve for the remaining unknown.

1

2-

2) Solving Systems by EliminationExample 1:

Solve the system by elimination.

4x + 3y = 4 Sub y = -2 into

4x + 3(-2) = 4

4x – 6 = 4

4x = 10

x = 2.5

Step 4: Use substitution to solve for the remaining unknown.

11

2) Solving Systems by Elimination

Example 1:

Solve the system by elimination.

Therefore, the solution is (2.5, -2).

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

x + 6y = 2

5x + 4y = 36{

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

x + 6y = 2 multiply by 5

5x + 4y = 36{1

2

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

x + 6y = 2 multiply by 5

5x + 4y = 36 becomes…

5x + 30y = 10

{1

2 1

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

x + 6y = 2 multiply by 5

5x + 4y = 36 becomes…

5x + 30y = 10

subtract

{1

2 1

1 2

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

5x + 30y = 10

5x + 4y = 36

1

2-

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

5x + 30y = 10

5x + 4y = 36

26y = -26

y = -1

1

2-

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

5x + 30y = 10

5x + 4y = 36

26y = -26

y = -1 Sub y = -1 in either equation.

1

2-

2) Solving Systems by Elimination

Example 2:

Solve the system by elimination.

5x + 4y = 36

5x + 4(-1) = 36

5x – 4 = 36

5x = 40

x = 8

2

2) Solving Systems by Elimination

Example 2:Solve the system by elimination.

x + 6y = 25x + 4y = 36

Therefore, the solution to the system is (8, -1).

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2) Solving Systems by Elimination

Example 2:Solve the system by elimination.

x + 6y = 25x + 4y = 36

Check: 8 + 6(-1) = 2 5(8) + 4(-1) = 36

2 = 2 36 = 36

{

2) Solving Systems by Elimination

Example 3:

Solve each system by elimination.

a) -3x + 5y = 7 b) 2x – 3y = 18

6x – 10y = -14 -2x + 3y = -6{ {

2) Solving Systems by Elimination

Example 3:

Solve each system by elimination.

a) -3x + 5y = 7 b) 2x – 3y = 18

6x – 10y = -14 -2x + 3y = -6{ {

Multiply by 2

2) Solving Systems by Elimination

Example 3:Solve each system by elimination.

a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6

0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12

{ {+ +

2) Solving Systems by Elimination

Example 3:Solve each system by elimination.

a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6

0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12

{ {

Always true.

The equations represent the same line.

The system is dependent.

There is an infinite number of solutions.

+ +

2) Solving Systems by Elimination

Example 3:Solve each system by elimination.

a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6

0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12

{ {

Never true.

The equations represent parallel lines.

The system is inconsistent.

There is no solution.

+ +

Always true.

The equations represent the same line.

The system is dependent.

There is an infinite number of solutions.

2) Solving Systems by Elimination

Example 3:Solve each system by elimination.

a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6

0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12

{ {

Never true.

The equations represent parallel lines.

The system is inconsistent.

There is no solution.

+ +

Always true.

The equations represent the same line.

The system is dependent.

There is an infinite number of solutions.

Homework p.128 #18-21, 33, 36, 46, 55, 56, 57, 62

Homework

p.128 #18-21, 33, 36, 46, 55, 56, 57, 62

Tomorrow: In-class assignment…come prepared!