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 non-linear equations The simplest linear system is one with two equations and two variables.

Graph of a Linear Equation

Graph of y = 3x – 2

x y

1 1

0 -2

-1 -5

Graph of a Linear Equation

Graph of y = –x – 6

x y

1 -7

0 -6

-1 -5

System of Linear Equations

Graph of y = 3x – 2 & y = –x – 6x y

1 1

0 -2

-1 -5

x y

1 -7

0 -6

-1 -5

System of Linear Equations

Graph of y = 3x – 2 & y = –x – 6x y

1 1

0 -2

-1 -5

x y

1 -7

0 -6

-1 -5

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 = –24x + y = 24

Choose 2nd equation: 4x + y = 24 Rewrite with single variable: y = 24 – 4xSubstitute into 1st equation: 2x – 3(24 – 4x) = –2

Solving Systems of Linear Equations

Substitution Method:2x – 3y = –24x + y = 24

Simplify: 2x – 72 + 12x = –2 14x – 72 = -2

14x = 70 x = 5

Solving Systems of Linear Equations

Substitution Method:2x – 3y = –24x + y = 24

Substitute x = 5 into either equation: 4x + y = 24 4(5) + y = 24 20 + y = 24y = 24 – 20y = 4

Solving Systems of Linear Equations

Substitution Method:2x – 3y = –24x + 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 = 11 -6 -6 x = 5

Solving Systems of Linear Equations

Addition or Elimination Method-easy:2x + y = 9

3x – y = 16

Add: 5x = 25Simplify: x = 5Substitute: 2(5) + y = 9

10 + 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 1st by 4: 8x – 4y = 36

8x – 4 y = 363x + 4y = –14

Solving Systems of Linear Equations

Addition or Elimination Method – medium:8x – 4 y = 36

3x + 4y = –14

Multiply 1st by 4: 8x – 4y = 36Add: 11x = 22Simplify: x = 2Substitute: 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 EquationsAddition or Elimination Method – hard:

4x – 3y = 25 –3x + 8y = 10

Multiply 1st by 3: 12x – 9y = 75Multiply 2nd by 4: -12x + 32y = 40

Solving Systems of Linear EquationsAddition or Elimination Method – hard:

12x – 9y = 75-12x + 32y = 40

Add: 23y = 115Simplify: y = 5Substitute (original equation) : 4x – 3y = 25

4x – 3(5) = 25 4x = 40x = 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