Systems of Linear Equations Block 44. System of Linear Equations A system of equations is a set or collection of equations that you deal with all together.

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Presentation transcript:

Systems of Linear Equations Block 44

System of Linear Equations A system of equations is a set or collection of equations that you deal with all together at once. Linear equations (ones that graph as straight lines) are simpler than nonlinear equations The simplest linear system is one with two equations and two variables.

Graph of a Linear Equation Graph of y = 3x – 2 xy

Graph of a Linear Equation Graph of y = –x – 6 xy

System of Linear Equations Graph of y = 3x – 2 & y = –x – 6 xy xy

System of Linear Equations Graph of y = 3x – 2 & y = –x – 6 xy xy Solution is (-1, -5)

Practice Solving Systems of Linear Equations Solve by Graphing the following systems of linear equations (see worksheet #1): #1

Practice Solving Systems of Linear Equations Solve by Graphing the following systems of linear equations (see worksheet #1): #2

Practice Solving Systems of Linear Equations Solve by Graphing the following systems of linear equations (see worksheet #1): #3

Practice Solving Systems of Linear Equations Solve by Graphing the following systems of linear equations (see worksheet #1): #4

Practice Solving Systems of Linear Equations Solve by Graphing the following systems of linear equations (see worksheet #1): #5

Solving Systems of Linear Equations Substitution Method: 2x – 3y = –2 4x + y = 24 Choose 2 nd equation: 4x + y = 24 Rewrite with single variable: y = 24 – 4x Substitute into 1 st equation: 2x – 3(24 – 4x) = –2

Solving Systems of Linear Equations Substitution Method: 2x – 3y = –2 4x + y = 24 Simplify: 2x – x = –2 14x – 72 = -2 14x = 70 x = 5

Solving Systems of Linear Equations Substitution Method: 2x – 3y = –2 4x + y = 24 Substitute x = 5 into either equation: 4x + y = 24 4(5) + y = y = 24 y = 24 – 20 y = 4

Solving Systems of Linear Equations Substitution Method: 2x – 3y = –2 4x + y = 24 The solution is the ordered pair (5, 4).

Practice Solving Systems of Linear Equations Solve by Substitution the following systems of linear equations (see worksheet #2): #1

Practice Solving Systems of Linear Equations Solve by Substitution the following systems of linear equations (see worksheet #2): #2

Practice Solving Systems of Linear Equations Solve by Substitution the following systems of linear equations (see worksheet #2): #3

Practice Solving Systems of Linear Equations Solve by Substitution the following systems of linear equations (see worksheet #2): #4

Practice Solving Systems of Linear Equations Solve by Substitution the following systems of linear equations (see worksheet #2): #5

Solving an Equation Addition or Elimination Method: Example: x + 6 = x = 5

Solving Systems of Linear Equations Addition or Elimination Method-easy: 2x + y = 9 3x – y = 16 Add: 5x = 25 Simplify: x = 5 Substitute: 2(5) + y = y = 9 y = -1

Solving Systems of Linear Equations Addition or Elimination Method - easy: 2x + y = 9 3x – y = 16 Solution is (5, -1)

Solving Systems of Linear Equations Addition or Elimination Method – medium: 2x – y = 9 3x + 4y = –14 Multiply 1 st by 4: 8x – 4y = 36 8x – 4 y = 36 3x + 4y = –14

Solving Systems of Linear Equations Addition or Elimination Method – medium: 8x – 4 y = 36 3x + 4y = –14 Multiply 1 st by 4: 8x – 4y = 36 Add: 11x = 22 Simplify: x = 2 Substitute: 2(2) – y = 9 4 – y = 9 -y = 5 or y = -5

Solving Systems of Linear Equations Addition or Elimination Method – medium: 2x – y = 9 3x + 4y = –14 Solution is (2, -5)

Solving Systems of Linear Equations Addition or Elimination Method – hard: 4x – 3y = 25 –3x + 8y = 10 Multiply 1 st by 3: 12x – 9y = 75 Multiply 2 nd by 4: -12x + 32y = 40

Solving Systems of Linear Equations Addition or Elimination Method – hard: 12x – 9y = x + 32y = 40 Add: 23y = 115 Simplify: y = 5 Substitute (original equation) : 4x – 3y = 25 4x – 3(5) = 25 4x = 40 x = 10 Solution is (10, 5)

Practice Solving Systems of Linear Equations Solve by Addition/Elimination the following systems of linear equations (see worksheet #3): #1

Practice Solving Systems of Linear Equations Solve by Addition/Elimination the following systems of linear equations (see worksheet #3): #2

Practice Solving Systems of Linear Equations Solve by Addition/Elimination the following systems of linear equations (see worksheet #3): #3

Practice Solving Systems of Linear Equations Solve by Addition/Elimination the following systems of linear equations (see worksheet #3): #4

Practice Solving Systems of Linear Equations Solve by Addition/Elimination the following systems of linear equations (see worksheet #3): #5